i like to fast play a flopped nut straight like this b/c a) if he has 2 pair or the dummy straight it is quite unlikely he is getting away from his hand, b) it actually disguises your hand b/c many players WOULD slowplay your hand, and c) a good number of cards can come off on the turn/river that kill your action.

i would bet roughly 2/3 of the pot on the river (given your flop/turn play), it may look like a busted draw of some sort and may garner a loose call. checking behind is out of the question b/c there is no reason to belive that you are behind.