Well it depends on the table, but I've noticed that
when you deviate from your general style of play for no other reason than to break the repetitiveness of it all
while you happen to be card dead then you are going to start
outplaying yourself.
If you're at a table where players are seeing 35-%40% flops yet the average pot is 5-10 BB's then you can rest assured that whoever calls your C-bet isn't doing so with A high K kicker most of the time. This means that the
best case scenario for you in this situation is that villain has at
least an under pair that they're reluctant to fold because they've waited so long for it.
It's more likely that they are strong though.
With that said,
if you must satisfy your curiosity, the
worst thing you can do in this situation is check the turn and call on the river because even if they were
bluffing, and calling with air, you'll be left with a decision after they've had a chance to improve for free. If they are in fact weak on the turn the only logical and possible way to find this out
with the least risk is to bet the turn.
If villain calls, you have your answer and if he folds then you have your answer.
This beats calling on the river because your
turn aggression gives you a
chance to take the pot AND get information while checking the turn and
calling the river
sacrifices your chance to take the pot with a potentially weaker hand, gives him a chance to
improve for free and has you paying the same price for information playing
passively on the river that you would have paid on the turn
aggressively, possibly stealing the pot.
What if he bets the river after your aggression on the turn you ask? Again,
you will have had your answer on the turn since villain called and it's time to let it go.
In the end, unless you have some type of read on your opponent, it's a good idea to let it go after your C-bet has been called and you have nothing but 2 overs or an under pair. If you do have a justified hunch that your opponent is weak and
you're willing to call the river then you need to bet the turn. This is what it means to play aggressive poker.
EDIT: If you're in position to call a C-bet with air then the only reason you should be calling a C-bet with the intention to steal on the turn is on a rainbow board/you have a read. Even then you will have more than likely been better off raising on the flop to avoid a scare card on the turn. Not a min raise, but 3x should do it. If you're called then you're probably beat, but if you're unsure and need to find out.
You've gotta bet it once again on the turn if you'd be willing to call a bet on the river. Same price most of the time and you'll be the aggressive. Best to let it go I say...be patient.