I'm trying to figure out who owns the Golden Claw so I can return it to complete the quest. I've killed the boss at the end and acquired the new shout, I just don't know who to return the claw to!
The claw belongs to Lucian Valerius:
you can find him in Riverwood at his shop, the Riverwood Trader, in the center of town:
In general, you can figure out what to do with a quest by tracking it in your quest journal: if you do that, you'll get a marker on your HUD's compass that'll lead you directly to where you need to go.
(Images from the UESP)