Your character must have either lived at Hogwarts or died at Hogwarts to in order to be able to play.
Example 1: Remus J. Lupin could either be played when he was a student, a teacher or the one who died at the Battle of Hogwarts. Example 2: Cornelius Fudge. He would be playable as a student but not as an adult or in the capacity of Minister of Magic. Example 3: Harry Potter is only playable as a student as are the children mentioned in the epilogue at the of HP#7.
If you are at a lose as to who to play, please see the HP Lexicon. It lists everyone.
As far as Founders go, they need to come in before The Hunger was trapped because it would just be too easy to trap The Hunger again.