Final Fight: Double Impact

UPDATE - From comments: "Just wanted to say that I just got this and you do NOT have to do the secret levels. I only did the normal levels."

Original solution:

It appears this needs to be done in a single playthrough. I did it in one sitting, just in case. It can be done in co-op and should unlock for both players.

Completed levels do NOT carry over, like they do in the Vault.

You need to complete all the levels including all the secret levels using less than 18 continues. Here are the secret levels you want (I borrowed most of this from the Vault guide):

Floor 28 - Crouch on the platform above the first pool of acid and slash the space in the middle.

Floor 40 - Crouch and slash the space above the ledge before the first pit of spikes.

Here are the secret door you do NOT want. Avoid these doors!

Floor 7 - When you enter the floor, walk left and hit the column. Warps to level 11.

Floor 14 - After the first swinging ball on a chain, jump up one step and crouch and slash the top step. The door will appear on the other side of the steps. Warps to 19.

Floor 17 - Slash the left side of the last arch. Warps to 20.

Floor 21 - Between the imprisoned (knight) and the exit, jump left from the elevated ground while slashing. Warps to 24.

Floor 34 - Jump and slash the left side of the the fourth pair of hanging pillars. Warps to 36.

Floor 39 - Slash the ledge immediately to the left of the first door after the pit of fire. Warps to 42.

Co-op makes this a bit easier. Only the boss fights should give you any real trouble. Even then, just throw in your magic attack once in a while (X+A) and they should go down. It will drain your life a bit, but it's probably a safe trade for the damage the bosses would do.

Upon completion of Floors 1-50 using less than 18 continues, the ending will play, and the achievement will unlock.