Possibly the hardest achievement in the game.
If you are trying to complete the game ASAP, try to get your first playthrough without dying, by just following the main path, and the 2nd to get all of the collectibles.
Take your time in the last level. It takes around 8 minutes to complete if done at a decent speed, but any mistake will make you restart completely. The other levels are a lot easier, and mistakes won't cost you that much
Also, make sure you note somewhere which levels you've complete without dying. There is no way to track it inside of the game, so you might have to redo everything once more if you aren't sure of it.
Besides that, I don't really have any tips... Doodle's a saver in most cases though.