The only two logical I can come up with are most damage dealt, or finishing blow.
However when I start attacking someone who is at 10% hp and kill them, I don't get the kill. So it can't be the latter.
When I attacked someone at full hp and got them down to 10%, and someone else finished them off, I still didn't get the kill. So it can't be most damage dealt either.
So what determines who gets the kill? It seems pretty inconsistent.