If your capping the turn, then why are you check raising the river? If you cap this turn you should be leading this river or check calling. Not check raising. (You seem to already know that based on your statements) Im assuming he just called your river c/r? Or else you would have included it in the hand history.

You are beat by AJ, AQ, AA - Ahead of AK, JJ. Given his PF action, I think we can rule out AJ and AQ, and likely JJ. This is where you have to make a decision does he have the AA or the AK.

I think bet/calling is the appropriate line on the river. I can't find a fold here though.