If you have played the first Bioshock you may have attempted this achievement. Unlike the first you can complete the game on any difficulty but you must not use the Vita-Chambers. Start on easy or normal and make your way through the game.
Take your time and avoid as much unnecessary conflict as possible. If you are overwhelmed either take cover or run away and use your health kits when needed.
Note: Once you start the game you can turn the Vita-Chambers off through the option menu. Doing this will stop you from making a mistake and using the chambers. Make sure you save on a regular basis.