Finish the game on any difficulty to net this achievement.
I would recommend to start on normal, then start a new game on Hardcore when you are finished with your character from your normal game.
After completing the Hardcore run you will net 2 more achievements after completing the campaign again, one for completing the game on Hardcore and for doing 2 playthroughs.
Also, after finishing your 2nd run on Hardcore, you will unlock Insanity Difficulty
This will help reduce the playthroughs you need if you are going for completion.