I check behind the flop, I call on the turn and reevaluate on the river.
If I check behind the flop and there is not bet on the turn, I bet myself.
As played, I would not bet the turn, since he might have easily king when he called you on the flop. You cannot fold river to $2 bet (into $6,45) and since you checked the turn, you should call the river (by checking behind, you also induced worse hands to bet which would fold to your turn bet).