EDIT - PLEASE READ THIS FIRST: I would like to thank everyone for giving such positive feedback, it's cool to know that this guide is still helping people. That said, I must ask that anyone having trouble with this achievement to NOT message me asking for help. I'm not trying to be a prick, but I haven't really played this game in like a year, I would just bring you down probably. And as I state later in this guide it is easier to do solo anyway. Just keep at it and don't give up, thanks again for all the support!
I suggest getting this achievement in co-op, preferably 2 players (that way there aren't tons of enemies cluttering the screen). This achievement sounds harder than it really is. Yes you only get one life, but if you've played through the game already and payed attention and took the time to figure out the patterns of the game, it's a piece of cake... or pizza I should say.
EDIT: After playing Survival Mode both solo and co-op, I've actually found that playing alone is easier, at least for me anyway. Try it on solo first and if you're still having trouble, try co-op. Just make sure one turtle attacks (bosses) at a time or things can get confusing.
The game shouldn't be too difficult on Normal. Beating up all the baddies is easy, the only trouble I predict people having is on the bosses. I've done my best to write out this guide for each boss based on my experience, and I can honestly say using these strategies I can beat every boss in the game without losing a life. And if I can do it, anyone can.
Level 1: Big Apple, 3AM
BOSS: Baxter Stockman
Baxter is one of, if not the easiest boss in the game. All you have to do is stay below him when he is flying so he can't shoot you, then when he lands get in close for a quick three hits with your regular attack (X). When his life is half gone he will pull out his second gun which is stronger and faster. Doesn't matter though just do the same thing as before, just make sure you hit him as soon as he lands on the ground and he won't be able to get a shot off.
Level 2: Alleycat Blues
Metalhead is not a difficult boss either. You have to be a little more careful, especially if you're trying to get multiple hits on him at once. I don't recommend that though. I suggest just go in and land one quick special attack (Y), get away, and go in for another. It's pretty easy since his attacks aren't very accurate or fast. Once in a while he'll do this jump kick (and he'll do 3 in a row every time), all you have to do then is just get away and let him miss 3 times.
Level 3: Sewer Surfin'
BOSS: No boss in Arcade version.
Level 4: B.C. 2500000000: Prehistoric Turtlesaurus
BOSS: Cement Man
It took me a few tries to figure out this bosses pattern, the main thing that kept catching me is when he would go underground. I used to try to jump away but he would get me every time. Then I figured out that if you just run away from him he can't catch, he's too slow. When he pops up from the ground don't go right in for attacks, wait. He'll throw a sludge ball at you. Dodge it and then go in for 2 or 3 quick hits, but don't stay close for too long because he goes back in the ground fast after being hit.
Level 5: A.D. 1530: Skull and Crossbones
BOSS: Tokka and Rahzar
Even though there are two bosses in this level it's VERY easy, even by yourself. I found that the easiest way to beat them is to just keep doing jump kicks. It does a pretty good amount of damage and it's fast. If you just keep jumping and kicking, staying in the air, Tokka and Rahzar can't even get a hit in. The only thing you have to look out for is Tokkas' razor shell spin attack. When he does this don't worry about attacking, just jump around dodging the attack until he stops.
Level 6: A.D. 1885: Bury my Shell at Wounded Knee
Leatherhead is a hit and move kind of boss, you can't really stay in too long or he'll pull out a knife and shank ya'. Get a hit on him and he'll do a charge attack. He'll go back and fourth across the screen 3 times (sometimes only 1 or 2, but rarely). Dodge them by jumping over him. When he stops he'll be on one of the far sides of the screen. Just make sure you're not on the same horizontal plain as him when he stops or he'll throw knives at you. Attack him from above or below, you can usually get 2 or 3 hits in before he goes for another charge attack. Just repeat this process.
Level 7: A.D. 2020: Neon Night-Riders
BOSS: Super Krang
Krang and his robot are actually pretty easy. All you do is get a few quick hits in, I usually get in about 3 or 4. After that move away, he'll usually go for a kick attack or a clap attack. Just dodge it quickly and go in for another 3 or 4 hits. That's all, he shouldn't give you too much trouble.
Level 8: A.D. 2100: Starbase
BOSS: Krang in a spaceship
This is one of the more annoying boss fights in the game. It's probably the one I have the most trouble with. Try to stay vertical with him, never attack him from the sides or he'll trap you in a bubble, which takes away damage. Try attack him vertically, when you see him shake for a second or two get out of the way. He'll try to slam you, just be quick and don't get greedy with the attacks, get what you can and dodge. When he starts to drop the little robots, take care of them fast. You don't want too many of those things walking around freely, they get in a lot of cheap damaging attacks. If you stay close to Krang you can take out the robots as soon as he drops them while getting some hits in on him. Be very patient with this boss, I suggest practicing in quick play.
Level 9: A.D. 1992: Technodrome: The Final Shell-Shock
Shredder can be a pain if you don't know his patterns, but once you do he's not so bad. Just try to work him into one of the upper corners so you can fight against the rail and keep the fight horizontal. Once he's in the corner go for a regular attack, but ONLY one. He'll dodge it by backing away into the corner. Once he does that back up a little bit because he'll go for an attack right after he dodges (usually a kick or sword swipe, sometimes a radial blast attack). After he misses quickly go in for 3 quick shots, he might block and attack once in a while but you can usually get some good hits in. After your 3 hits back away and repeat this process. Let him attack and miss, go in for 3 shots and back away. When his health gets low he'll do this attack where he levitates and tries to shoot you with a turtle soup beam (similar to the 1989 arcade). However this attack is very easy to avoid. Just stay close to him while he's in the air (almost right underneath him) and wait for him to land to get a couple of free shots.
Well that's it. I hope this guide helps anyone having trouble with those frustrating boss battles. The only other advice I can give is to help out your team mate. If your going for the achievement that doesn't mean you can let your friend do all the work, remember you BOTH only have one life. Help each other out. Most boss fights usually have a pizza or two in them. Use this to your advantage, attack the boss for a while and let your friend sit back and retain their health. If you start getting into trouble and losing health tell your friend to jump in and fight for a while so you can get the pizza, then if your friend starts getting into trouble you can jump in and help them. Just be smart about how you distribute the pizzas, don't grab one if you don't need one and be careful fighting near one because you can get knocked onto one accidentally.
Please rate and tell me if there is anything I can add or fix. Good luck!