This is a multi-step achievement, not a difficulty-specific achievement (difficulty options do not exist in this game).
To complete the game the hard way, you have to complete the following steps:
Mines Level 2 or 3 or 4: obtain the Udjat Eye. To do so, find the golden key and take it to the treasure chest with the golden lock (this step is not necessary, but it helps a lot).
Jungle Level 1 or 2 or 3: find the entrance to the Black Market. To find it, the Udjat Eye will flash/click when you are close to the entrance. The faster it flashes/clicks, the closer you are. Use bombs or a Mattock to get to it (the Black Market can also appear in the Haunted Castle, if you enter the Haunted Castle from Jungle Level 1 or 2 only. If you go to the Worm, the Black Market will never appear, since you are taken straight to Jungle Level 4 after you the exit the Worm).
Black Market: buy or steal the Ankh ($50000 if you want to buy it). Once you have it, do not die until you get to the level with the big blue stone head in the Ice Caves, or you'll have to start all over.
Ice Caves Level 2 or 3: when you see the big blue stone head, die in anyway you want (except getting crushed by falling platforms or shattered after being frozen by a Mammoth. It might not work). You will then spawn inside the stone head and you will obtain the Hedjet helmet (without this, you cannot open the golden door to the City of Gold). You will also keep your gear except handheld items, such as Mattocks or Shotguns (if you're playing Co-op, the player who does not have the Ankh can bring handheld items with them. Once the player with the Ankh dies, both players will spawn inside the stone head, and the other player will still be holding their handheld item).
*Alternate route to get the Hedjet: if you go to the Worm in Ice Caves Level 1, you can still get the Hedjet. After the Worm, you'll have to enter the Mothership and escape, then the blue stone head will appear in the next Ice Caves level.
Temple Level 1: find and kill Anubis (he shoots pink circles at you). Takes 2 bombs to kill him. After you kill him, pick up the Sceptre that he drops.
Temple Level 2: take the Sceptre to the golden locked door to the City of Gold (make sure you have the Hedjet helmet as well).
City of Gold: find and take the Book of the Dead. It is always in the center of the level. You will need bombs or a Mattock to get to it. After you pick it up, Anubis II will appear and try to kill you. He summons red skeletons. You can kill him here or leave the level, but you will have to fight him in the next level if you do so. Two bombs to kill him.
Olmec: when the Book of the Dead starts laughing really fast, you are above the entrance to Hell. You need Olmec to fall in the lava right below the entrance to Hell, then land on his head and go inside the entrance. Be careful, Magma Men can jump out of the lava and try to knock you off into the lava.
Hell: survive the 3 levels.
King Yama (final boss): beat him and escape to the exit in the upper right or left corner. The way I beat him was by destroying his hands first with sticky bombs (1 bomb each). Then throw sticky bombs at his head (4 bombs). Also, if you have a Shotgun, throw a rope under the brick structure in either the upper left or upper right corner. Climb to the top of the rope and shoot Yama continuously from there. Yama will not be able to attack you while you are under the brick structure. If you need to throw a rope higher, use sticky bombs to destroy part of the brick structure.
Some tips regarding King Yama:
1. Don't get too close to his hands, they can punch you.
2. If you climb up high enough (to the point where you're right below the brick structures or at the very top), his hands will rise and slam, causing skulls to fall on you. That was the first way I died fighting him.
3. After you destroy his hands, his head will come off and move, from left to right (or right to left). Again, if you're close enough to the brick structures or higher, his head will slam the ceiling really fast and drop skulls on you.
4. There will be enemies trying to knock you off the chains. Be careful.
5. I recommend destroying the spikes near the chains. I died at least twice by them.
6. At the very beginning of the fight, Yama's two bodyguards will appear, at the left and right of you. They drop bomb boxes, so in case you're low on bombs, I recommend killing them ASAP. If they get too close to you, they will melee you and you will lose 2 hearts. They can also throw bombs. Use a Shotgun, Freeze Ray, or Spike Boots to kill them if you can. That way their bomb boxes fall safely on the floor, instead of the lava.
***The title update that came out in December 2012 fixed the Xbox freezing glitch for spending more than 99 minutes to beat the game. You should have unlimited time to get any of the achievements except Speedlunky.