Dante Must Die mode is the hardest difficulty mode in Devil May Cry 3 so as you can guess, it's really, really, REALLY hard and arguably the hardest DMD in all of the Devil May Cry series. You unlock it after beating Hard mode so you can skip over Very Hard if you want. If you do beat Very Hard, you can unlock Super Dante which makes it a bit easier but it's still quite a pain in the ass.
Enemies in Dante Must Die mode do a lot of damage, especially bosses, and they can Devil Trigger (the enemies, not bosses) which makes them harder to stagger but it's not impossible to do so. Enemy placements are the same as in Very Hard mode, meaning you'll get stronger enemies very early in the game.
Before tackling Dante Must Die mode, I would highly recommend maxing out your Small and Large Vital Stars as well as your Holy Waters. Holy Waters are a freaking godsend in this game and although they don't kill enemies instantly, they weaken them very considerably which can be very useful against groups of enemies you're not that great against. I found the best way to gather red orbs quickly is to do Mission 20 over and over again on Easy difficulty. If you can S rank it, you get 14,500 red orbs and it usually only takes around a minute to kill Vergil if you know what you're doing so you can get around 29,000 red orbs every couple of minutes. I managed to get 400,000 red orbs in an hour.
Of course having a max health bar and max Devil Trigger bar is recommended as well.
The Kalina Ann is a wonderful gun to use in DMD mode as you can use it for crowd control if there are too many enemies on the screen. If too many enemies get too close to you, use the Kalina Ann so knock them all back and give yourself some breathing room to attack.
Also I can't stress this enough: DO NOT FIGHT BATTLES THAT YOU DO NOT NEED TO FIGHT! If the door you need to go through isn't red, ignore all the enemies in the room and just get out of there to save yourself the hassle unless you're in dire need for health.
Mission 1
-There's nothing special to this mission. I believe a Wrath spawns in this level now but other than that, it's the same basic enemies from before.
Mission 2
-Again the basic enemies are here though you'll also fight Sloths here instead of Mission 3. The Hell Vanguard now disappears and attacks you three consecutive times and if he manages to hit you, it takes off quite a bit of damage so make sure to pay attention.
Mission 3
-This is the first mission in which you'll encounter Enigmas. If you ever fight Enigmas, I would highly recommend taking them out first as they're one of the most annoying enemies to kill while they're Devil Triggered.
Cerberus has no new attacks at all but the main difference about him is that he gives about a one second warning before he's ready to try to tackle you so you need to be paying attention and be ready to roll out of the way. If you get caught in that tackle, you'll easily lose about 1/4 to nearly a half of your first health bar (assuming you're not doing this as Super Dante.)
Mission 4
-The only real threat of this mission are the Enigmas at the beginning of the mission. As usual, make sure to take them out ASAP so that their arrows don't become a problem.
Gigapede really loves to use those purple electricity orbs constantly so if you're on its back, make sure to pay attention to when his body starts forming those orbs so you can jump off and do some hardcore dodging.
Mission 5
-The Blood-Goyles are a pain in the ass to deal with, mainly because they'll usually dive at you when you're not paying attention to the others while you're constantly trying to get the one you're focused one into stone. However a weapon to use for Blood-Goyles is the Kalina Ann as it will put the Blood-Goyles into stone with one hit.
Jester isn't that bad. It takes longer to stun him so make sure to mash the X button and have Ebony and Ivory equipped. Make sure to stay on the very outside of the arena as it's much easier to dodge the energy balls this way.
Agni and Rudra are probably the hardest bosses in Dante Must Die mode. Their attacks are just so powerful and since you need to kill two bosses instead of one, you're going to be doing a lot of dodging and careful attacking. Your best bet to fight this fight efficiently is to try to knock their swords out of their hands which leaves them wide open to attacks. In order to do this, as they get ready to strike you, you need to strike them right before their swords hit you. I would recommend practicing this on easy before attempting this fight. It will save you a lot of healing items and probably save you some money from buying a new controller.
Mission 6
-This is probably the easiest mission in Dante Must Die mode. You don't need to do all three trials. Just do the Trial of Wisdom and Skill and you're already done.
Mission 7
-This level introduces the Hell Greeds, which are really, really deadly on Dante Must Die mode due to their ability to spawn other monsters to attack you. Using at least one Holy Water isn't a bad idea to weaken them but it's not necessary. You just want to kill them as fast as possible before the room gets filled up with too many demons. Also make sure to keep an eye on when they're about to swing their coffins so you don't get knocked back and lose health.
Vergil isn't actually any harder than he is on the other difficulty levels. He just takes longer to kill and he has one new attack. He will now use his Summon Swords which aren't that hard to dodge as long as you keep moving. Using Trickster is recommended for the Vergil fights.
Mission 8
-The Envies in this level can get rather tedious if they get bunched up together and gang up on you. The Kalina Ann is a good weapon for this level as it'll knock them back so you can get some breathing room.
The Envies make the Heart of Leviathan a nasty battle. Make sure to focus on the right lung as you'll want all the Devil Trigger you can get. If the Envies get too close, spam the Kalina Ann to make them stagger backwards before continuing to attack the lung. Once it collapses the Heart will reveal itself so use your Devil Trigger and just spam your combos on it. Usually after the first time it opens it'll start shooting those red lasers after it reveals itself again so make sure to jump over them.
Mission 9
-The Arachnes are introduced in this level and while they're not hard, it takes a while to defeat all of them so try to focus all of your attacks on one at a time and don't be hesitant to use Devil Trigger at all to aid you.
Nevan has no new attacks whatsoever. All she really does is just attack more before she starts taunting you. Once she starts laughing at you, that's when you start spamming combos on her to make her vulnerable to damage.
Mission 10
-The only dangerous thing about this mission are the Arachnes near the end of the mission. This is a pretty short mission.
Mission 11
-Beowulf is still a hard boss but he doesn't have any new attacks but one of his attacks is a little bit deadlier now. During his second phase, whenever he slams his fist into the ground and a giant light beam surrounds him, a cage will fall from the sky and the light beam will knock the cage in your direction. You can either dodge it or if you're quick with your reflexes, you can knock it back.
Whenever he starts making cages fall down after the first phase, make sure to use the Swordmaster style and hold RB+Forward and then push B to throw your sword at the cages as they come toward you. They'll get knocked back at Beowulf, damaging him and they'll drop health as well if you need it. He usually hits around 4-6 cages toward you on DMD.
Mission 12
-You'll want to have a couple of healing items for the mission as your health will constantly deplete. All the enemies in this level drop health but considering you're playing on DMD mode and they will Devil Trigger at some point, it will take longer to kill all of them, especially when you get to the next to last room containing a Hell Vanguard. It's annoying to have a bunch of Hell Lusts charge at you while you're fighting off an even more dangerous Hell Vanguard so don't hesitate to use a healing item if you need to.
Geryon has no new attacks at all but during the first part of the fight, he'll launch missiles at you while he charges each time until you get to the second phase of the fight. It just takes longer to kill him and his Quicksilver orbs chase you around a lot faster than before. Other than that he's not all that bad and isn't that dangerous.
Mission 13
-Make sure to bring Beowulf for this level as you'll want it against Vergil. Vergil's only "new" attack is the Summon Swords attack and since you don't damage him as much now, his Devil Trigger is incredibly frustrating as he can regenerate his health to full rather quickly.
As usual, attack him once he uses the Killer Bee attack (the drop kick attack).. Once he goes into Devil Trigger himself, equip Beowulf and use Killer Bees or Straights of your own. Three of either of those attacks should knock him out of Devil Trigger quickly so he won't regenerate that much health. However if he has Summon Swords out while in Devil Trigger, DO NOT attack him until those swords are gone. You'll take a lot of damage if you attempt to get him out of Devil Trigger while those swords are out.
Mission 14
-The Hell Vanguard fight during the Temperance Wagon sequence is going to be a massive pain because you have close to no room to navigate at all when the Hell Vanguard decides to pop in and out of midair. Take down the lesser enemies first before focusing on him.
The Hell's Highway area of this mission is the hardest part. The pawns move extremely quick and attack much faster than the other difficulty levels and the Knights make it a point to constantly jump up and down, giving you little time to attack. Since you can't really get the chess pieces separated at all, it's best to just attack what you can.
Mission 15
-The Fallens are huge pains. The Kalina Ann is great for stunning them if you hit them twice with it. Use Helm Splitters (jump in the air and then push Y to come down with your sword sticking out) to break through their shields and then wait for them to start attacking before distancing yourself and using the Kalina Ann.
Make sure to avoid all of the optional Arachne fights in this level. There are two you can skip before there's a mandatory fight near the end against them.
Mission 16
-At the start of the mission, if you go through the left door first, make sure to go DOWN the stairs, not up. If you go through the door at the top of the stairs, you'll be forced to fight some Hell Greeds which only puts you at risk of losing health before the boss fight in this level.
With that said, also avoid the fights along the stairwells. There's no need to fight the enemies here.
Lady isn't hard at all and she seems to be the weakest of all the bosses unless you get smacked directly in the face by all her missiles from the Kalina Ann. The best way to take on this fight is to hit her three times while she's setting up her Kalina Ann and then hide behind the stone pillars so that she can't see you. She'll start winding up the Kalina Ann so come out, hit her three times and hide again. Rinse and repeat.
Mission 17
-The only real dangerous part of this mission is the fight against the four Abyss's near the end of the mission. Just remember that if you knock them back with a Stinger, they'll come charging right back at you as they'll land right on their hands and knees. This is something to remember since when they Devil Trigger, chances are you won't knock them out of their charge attack if you try to Stinger them again.
Doppelganger is still pretty easy. It just takes more hits to open the light windows. Don't get frustrated if he keeps shutting them. You can get him into a loop to prevent it. He'll usually charge at you after taunting you. When this happens make sure to leave the light window closed and make sure it only takes one hit to open it. When you open it, he'll be stunned. Quickly run to the next window and hit it again until he charges at you. Jump to avoid his attack when he gets near you and then continue hitting it when you land. It'll stun him again and he'll start charging at you again. He'll keep doing this so take advantage of it.
Mission 18
-This is easily the hardest mission in DMD mode. It's not because of the bosses you have to fight again. It's because of that damn chess board right at the beginning of the mission. Chances are you won't be able to kill all the pawns before all of the other pieces in the back start moving so you'll need to constantly keep an eye on the Queen as well as the Knights jumping around, the Pawns trying to hit you and the Bishops shooting fire, as well as the Rooks and their lasers. The worst part is the chess pieces can be revived at any time during this fight so you'll want to take out the King ASAP. Holy Waters are great for weakening up the chess pieces. I would finish off the Bishops first after the Pawns as they have the ability to heal. Then take out the Knights then the Queen, then the Rooks and finally the King. Ignore any chess piece that gets revived unless it's a Bishop.
In order to advance to the end of the mission, you need to form a circle on the boss slate. The quickest way to do this is to defeat Cerberus (light blue), Agni and Rudra (green), and then Beowulf (white). These three will form a small circle and the portal to the exit will appear.
Mission 19
-During the third fight against the Abyss's (the one with the giant hour glass in the middle of the arena), save yourself the time and hassle and use three Holy Waters to kill off the first three Abyss's and then use your weapons to finish off the last two that appear. It's really, really hard to kill all five before time runs out and it would just be a waste to lose health and start over again.
Arkham is really dangerous in DMD mode. During the first phase of the fight, as you attack him, keep a constant eye on his arms as he'll swipe them at you from time to time. Roll or jump to dodge this. As you keep hitting him, he'll start turning yellow eventually. Back up and run away when he does this and either Trickster to the side or roll to the side to avoid the things he shoots from his back at you. During the second half of the fight, Devil Trigger is not available to you so you'll need to be extra careful this time around. The same strategy applies here; attack him and dodge his arm swipes and back off when he turns yellow. However this time if you're not fast enough to get back, he'll grab you and will drain your life very quickly. You'll be in control of Vergil and in order to stop the life drainage, you need to hit him once. Don't hesitate to use healing items as you're near the end of the game.
Mission 20
-As usual, Vergil's only "new" move is the Summon Swords attack. However he seems to like to go into Devil Trigger and then use his Summon Swords attack this time around when he's low on health so this fight will last a lot longer than the other two. Also he'll use three sets of 4-5 Helm Splitters this time around instead of two so you'll want to constantly keep moving when he's in that phase of the fight. Whenever he goes into Devil Trigger, equip Beowulf and use Killer Bee's on him or Straights until he gets out of Devil Trigger and then continue your strategy on him.
It's also worth noting that when he gets in the crouching position to throw those three swords at you (the ones that form black orbs), you can Stinger him after he throws the first one to knock him out and stagger him, leaving him open for an additional 7-8 hits. Take advantage of this. Each time he uses a Helm breaker as a counter attack, he'll use a Stinger afterward so roll to the side and attack him. He'll be vulnerable for 7-8 hits here as well. And finally after his surge of Helm Splitters, once he gets out of Devil Trigger, he's open for more attacks. Take advantage of these openings and you'll slowly dwindle his health down and make sure to get him out of Devil Trigger as soon as possible. It's very annoying when he uses Summon Swords after using Devil Trigger but as long as you keep up your barrage of attacks you'll eventually win.
For beating Dante Must Die mode, you'll unlock Heaven or Hell mode. Everything dies in one hit, including you, in this mode so it's a good mode for some sweet, sweet revenge if you hated the enemies and bosses in DMD. You also unlock Super Sparda who has infinite DT AND his health regenerates. Super Sparda can also use the Doppelganger and Quicksilver styles for an infinite amount of time.
Happy hunting and good luck! You're going to need it.