I've seen speculation here that the king Kvothe kills is Ambrose, which seems like the logical choice. However, it almost seems too logical. I think that Kvothe kills someone (perhaps Baron Jakis) and Ambrose ends up as king because of it. Thus, Ambrose is the king in the time the story is being told. That would help to explain the huge reward for Kvothe's life. It would also explain why Kvothe is trying to keep his friends from joining the King's army.