I've been thinking about this for some time and I think I now know who he is.
Harrison is a Klingon.
but he is a Klingon affected by the Augment Virus and/or has been altered to look human a la "Trouble with Tribbles". If he's an augment, then we have our link to Khan.
This explains why part of the movie is set on Q'onos. It also explains why he's referred to as Starfleet's best agent. They mean Double Agent.
It also explains why Kirk refers to the beginning of an all out war. He means with the Klingons.
The scenes where Harrison is fighting/killing Klingons is because even the Klingon Empire want him dead - either because he's an augment/double agent/insane.
What do you all think?
P.S The reason the Klingons wear helmets it because they are augments too. A way for normal Klingons to set them apart.