In order to get this achievement you need to
-Complete all Light Worlds from 1-7 and get an A+ on all of them
-Complete the Dark World for all of them (A+ is not needed here)
-Collect all 100 bandages
-Find and complete all Warp Zones (including all character warp zones)
The negative levels and Teh Internets do not have to be completed.
For a complete list of all bandage and warp zone locations I found a great guide which was made by Peter Eykemanshttp://www.diygamer.com/2010/10/super-meat-boy-bandage-warp-...
For video guide of someone completing the levels I found a great playlist of a walkthrough by MahaloVideoGameshttp://www.youtube.com/user/MahaloVideoGames#grid/user/07B70...
You can check your completion percentage on the Main Menu by going to Statistics.
Tips: You will need a lot of patience for this very difficult achievement. Unlocking characters also helps for some levels. The characters I found myself mostly using was Meat Boy(obviously), The Kid, and Tim. I would recommend to get the bandage and A+ on separate runs.
*For the last world(Cotton Alley) on level 7-12X there is a bandage that is technically a glitch. The bandage is not counted in the game and can result in you having 101 bandages. Since getting this bandage can result in you having having higher than 100% completion you can get this bandage and not do 5 Dark World levels in the Cotton Alley.
*This method worked for me but I cannot be sure if this was an isolated achievement glitch as it still says I have 99% completion.