You can complete all difficulty levels once you have completed the game by just completing the last chapter on each Act.
This means you must do 6 chapters on Hard, one of them being the final boss battle.
All but Act 2, maybe Act 3 and Act 6 are easy to complete on Hard if you have completed Normal.
Some people say you must complete the very first chapter on Hard, but I didn't need to, but if you want to be sure you can.
An important thing to note is they must be done in Order, if you do any chapter out of order the achievement WILL NOT unlock. Some claim that the order does not affect unlocking, but I have heard more than a few people state the order is important so I would not risk playing missions out of order.
The same applies to Casual and Normal if you need them. But you must have completed the game once to do this trick so you most likely need Casual and Hard.