Main Quest: must kill Draugur overlord
Companions: must kill hargravens
Dark Brotherhood: need I say more
Thief: Must kill Mercer faye
Bard: First quest you have to kill someone.
Collage of winterhold: many, many people you have to kill
Civil War: must take cities and castles by yourself.
Daedric artefacts: they're daedra what do you expect
If you take out all that plus most other miscellaneous quest require someone's death, then there really isn't must to do.