In fact, everything is quite simple and at the same time difficult. The main task is to choose the right house, since each of the four will have one unique story quest available. For example, students of Gryffindor will complete the task of Nearly Headless Nick.
To get to Azkabna, you must select Hufflepuff faculty at the start of the game. Only him. You won't be able to change the faculty during the game!