$2 NLHE Full Ring: Over Pair on a double paired board

I have been playing time and had chipped up a little bit. The Villain has played 8 of the 10 hands so far after sitting down. What is the best play?

Full Tilt No-Limit Hold'em, $0.02 BB (8 handed) - Full Tilt Converter Tool from http://poker-tools.flopturnriver.com/Hand-Converter.php

CO ($1.24)
Button ($3.63)
SB ($4.38)
BB ($1.82)
Hero (UTG) ($2.52)
UTG+1 ($1.06)
MP1 ($0.80)
MP2 ($2.42)

Preflop: Hero is UTG with K , K
Hero raises to $0.06, 2 folds, MP2 calls $0.06, CO calls $0.06, 3 folds

Flop: ($0.21) Q , 3 , 3 (3 players)
Hero bets $0.16, 1 fold, CO calls $0.16

Turn: ($0.53) Q (2 players)
Hero checks, CO checks

River: ($0.53) 6 (2 players)
Hero checks, CO bets $0.69, Hero calls $0.69

Total pot: $1.91 | Rake: $0.09

Anybody? Any views?

This looks good. It's a great spot to slow play. I'd be betting the turn, but the river should almost certainly a check.

I think you played it well.

If you bet the turn and he calls, you will be scared to play on the river and your opponent can bluff you easily.

If you bet the turn you will get paid only if your opponent has A (and probably he folds), if he has Full house or a pocket hand. it will be difficult to value this hand.

You checked twice on the turn and the river and your opponent think you are weak here. good call, you are favorite, but pot control here is nessecery, it's a dangerous action.