What line should I take here? I've TPTK but board is very wet. Should I've betted more on the flop and turn? Can I bet on this river? If hero checks what to do against a bet, fold? I'm thinking on the flop and turn I'm ahead but the river beats my TP.

Poker Stars, $0.02/$0.05 No Limit Hold'em Cash, 6 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

Hero (BB): $5.92 (118.4 bb)
UTG: $1.58 (31.6 bb)
MP: $4.41 (88.2 bb)
CO: $2.48 (49.6 bb)
BTN: $5 (100 bb)
SB: $3.42 (68.4 bb)

Preflop: Hero is BB with Q A
3 folds, BTN raises to $0.20, SB folds, Hero raises to $0.60, BTN calls $0.40

Flop: ($1.22) 5 7 6 (2 players)
Hero bets $0.70, BTN calls $0.70

Turn: ($2.62) Q (2 players)
Hero bets $1.60, BTN calls $1.60

River: ($5.82) K (2 players)
Hero ???

Pre is fine. Then on the flop I´m, not cbetting OOP because Imo if i give him a range of (we have no stats right ?) JJ-88,ATs+,KJs+,QJs,AQo+,KQo.....of course he can 4-bet AK. I thinkg that no better hand folds and not much worse hands are calling there of course he can peel one more card let´s say with KQ but what is you plan on next streets ? I´ll just try to play more pot control because with no hand (against his pre calling range you have 50% equity pre and on flop about 45%. Big pots are for the big hands and this is not the case and we want to KQ, KJ stay n the pot and if we are not expecting him to cbet with air we can just check. Of course it is villian dependent. The turn is fine but we inflated the pot I don´t expect him to call with air or some draw. Then river is small bet like 0.50 and if raise you just fold. Imo by this small bet you can force him go to showdoswn with hands that you can possibly beat.

Just call pre-flop. AQo is too hard to play OOP. As played, the 3bet doesn't win it right away and starts creating pot-size problems, which you compound on the flop and turn.

Then you could have c/f on flop (if villain bets) and got away cheaply and wait for a better situation. This is why position is so critical.

Very hard to put villain on a range since he was button and 3 folds. Could be stealing with anything, therefore all straight draws are a possibility, including hands with a pair and draw like 8-7 or 8-6 and hands with a pair and overcard like A-6 or K-5.

As played, check the river and hope he has paired one of the flop cards (he should check behind hoping that you have just given up) or on a pure draw (you may have value as a bluff catcher).

You haven't said anything about the villain. Is he a reg, loose, weak, unknown?

That should affect your preflop decision (3bet or not), your flop (cbet or not), turn is obvious bet as you played the previous streets (maybe a bit bigger), your river (not betting obv but does he bluff missed draws). Generally as said, this kind of flop is not very good for cbetting and playing the whole hand oop.