AK & other big hands like AQ are commonly misplayed post-flop. They will be the best hand pre-flop just as often not, depending on the number of players dealt in. They will usually need improvement by showdown, therefore considered drawing hands. With a pair on the flop good straight forward play will get the money.
What about when Ace King or Ace Queen miss the flop? This creates a problem for most players in limit holdem since they will miss the flop 68% of the time. There are easy to use guidelines when the hand misses the flop that I have used for years. The following are all assuming the hand was raised with preflop. Also the hand did not improve by making a pair and has no drawing potential such as a flush or gut shot straight draw with two over cards. Here is the correct strategy on a flush or straight draw with hands like AK.
On a short handed flop with two or less opponents a continuation bet should be made no matter what flops as long as you where not re-raised preflop. Ace high will still be the best hand many times. If not you at least have to over cards to the flop.
By betting on the flop you will be able to better gauge an opponent if they show strength with a raise or check raise. If an opponent shows strength keep in mind that if they have a pair, two over cards will only win about 25% of the time. If they have no pair and a draw they are still about 50% to win.
Playing Ace King High with no pair can be difficult since many sophisticated players will automatically try to steal against a pre-flop raiser when low cards fall. But most players usually have at least a draw or pair when they bet or raise. To fold is usually correct depending on the opponent. And even more so if out of position.But if you want the table to know you cannot be manipulated or ran over.
You can bite the bullet and make a stand at the cost of a bet or two. This may actually be better in the long run since players will think twice about trying to take pots from you in the future. If you are not playing with these opponents for long and the pot is small, to fold is correct when someone shows strength.
If no one shows strength, a bet may win the pot right away. If not should be followed by another bet on the turn since many players will automatically call a bet on the flop and then fold on the turn if they do not improve.
The only time checking the turn would be correct is against a tight passive player who is not likely to bluff and will usually only continue with a hand of strength.
On the river the correct play will vary depending on position. If you have position on your opponent at the river, a bet will usually only be called by a superior hand like a pair. So the hand should be checked down unless a bet has any decent chance of getting an opponent to fold a superior hand.
If out of position this is a tough spot in holdem. The only thing that can be done is to pick the best of two choices that are not very profitable. I always tell myself, if I'm stuck with no obvious correct play, choose the aggressive one. So bet once more, this will keep the pressure on your opponent and not on you. If you don't win the pot you will at least have guaranteed yourself action on future hands.
This is a much easier spot. If you not improved to a pair, flush or straight draw. It should be checked and folded. This is tough to do when there is this nice pot just sitting there. It is the best play though since it is likely not the best hand anymore. And if it does improve on later betting rounds, will often make second best hand. Betting or calling here is like throwing money out the window. If a draw is present however continuing with an aggressive approach is definitely best. Check out the betting poker hands section for this.