You have to start a New Game and finish all levels without quitting AND not die more than 5 times.
This is what I did for the Xbox 360 and Xbox One versions and will be doing this one the same way. (I'm trying to find a way to use a controller on my Win 7 machine, but no luck so far... using the keyboard sucks!)
Finish the game normally, dying as many times as you want. Then keep practicing the later levels gradually from end to middle. As in, start with the last level, practice it a few times to get it perfect. Then start the next to last level and finish the game from there to get it perfect. Then keep going backward each level and finishing the game from there. You'll get competent enough to where you can start a new game and complete the game with maybe a few deaths, but as long as you have 5 or less, you're good!
Most people usually leave this achievement for last, after getting all the hidden eggs and such. You do not need any hidden eggs to get this achievement.
video credit: YouTube user
JuanArekHere are my achievement lists for Xbox 360 and One to show I did get this one twice:
LIMBO on Xbox 360 - completed on Feb 9, 2013
LIMBO on Xbox One - completed on Apr 22, 2015