could i have played this hand any better?

[buddhablessed1]

online full 10/20 texas holdem game. i'm on the button with AKs. the action goes limp, limp, limp, raise, call, i decide to 3 bet in order to drive out the blinds and force the limpers to pay the maximum or get out and get some dead money in the pot. the blinds fold as does two out of the three limpers, everyone else calls. flop comes J75 with one of my suit. everyone checks to me i bet hoping to get a free card on turn and to further define my opponents' hands. limper folds, original raiser calls as does cold caller. turn is a 3 not of my suit making a complete rainbow. both opponents check to me again and i decide to bet again because no one has shown any strength, a few hands earlier i had played KK in the same fashion and won a big pot, and if i felt any resistence via a raise i'd get out. both opponents call again leaving me zero clue with where i stand. river pairs the J, a good card for me. both check to me and i decide to showdown the AK thinking no one is gonna fold for one more bet on a pot this big if they can beat A high. first raiser mucks, i get excited, cold caller turns over 22 to take down the pot with an awful two pair.

my question is: did i overplay my AKs on any street. would it have been more correct to just call preflop and let the field in and hope i make a flush? my reason for 3 betting is two fold, i could have the best hand and if i didn't but flopped top pair i wanted a better chance of it holding up by trimming the field. would anyone play this hand any differently?