There's been some terrible advice in this thread so far.
Your preflop raise is fine, Normally I make it 4xBB + 1BB for every limper, so that would be $3, but $2.50 is fine.
When he reraises, you need to call another $3, so you easyily have odds
to setmine, plus with position, you can sometimes win the pot withough flopping a set.
The flop is interesting. If he just cbet, it would be an easy fold. His check can mean a few different things. IMO, these are teh 3 most likely explinations: 1) he could be slowplaying quite a few hands (AA/KK/TT/AK) 2) he could have made a play preflop realised he got called and given up. 3) He could have QQ/JJ type hand and realise theres not point betting this flop.
In all 3 spots, checking the flop is best. If he has a big hand, obv you dont want to bet. If he has missed, we're ahead, so no real need to bet, as he'll just fold. If he has QQ/JJ, he's probably going to call the flop bet.
On the turn, you have to presume you are beat, and even if he's not slowplaying, getting him to fold his mid-stregnth hand is too risky.
I like your reasoning on the river...his hand certainly looks like JJ (as it happens, JJ and AQ have pretty much the exact same value on this board). However, 1) you presumed you could push him off it...a big mistake to presume such a thing readless at these levels and 2) you forget what your hand looks like to villain. You look like you have a weak hand,and so I would just give up and check if I were you.