Fold preflop.

Checking behind and betting the river when checked to credibly represents a pot control play with a hand like A-Q or K-Q, and avoids a check-raise when he has a hand. If he checks to you twice, he probably only has a really weak bluff-catcher and won't call often at all. I'm not sure if he's thinking this way, though, so maybe you should just semi-bluff the turn.