This achievement is a little tricky.
To make things easy, make sure you have 4 people total playing this game. You will need to help each other out in order to get through. Also, take note that everyone has 20 lives and only 20 lives.
Pizzas don't come up very often, so healing yourself will be tricky. Don't snag pizzas if you don't need them, but be careful: it is possible to get knocked into a pizza unwillingly.
Now, the achievement doesn't require you to just beat the game over Live. You have to beat the game ans still be alive after Shredder is defeated. This means if you ran out of lives and everyone else didn't and they beat the boss you won't get the achievement.
To save lives, try going for the other achievements while doing this. What I mean is when you fight the first boss, try to do it without dying. Try dodging all of the ice traps, don't get lit on fire when saving Splinter, and try taking care of the third to last boss w/o losing 2 lives.