This is an interesting question. The short answer is that Hamlet would be a thoughtful king; this could be good or bad. Throughout the play, Hamlet thinks things through. It would not be out of order to say that Hamlet over-thinks things. As Hamlet proceeds to avenge his father, he...
This is an interesting question. The short answer is that Hamlet would be a thoughtful king; this could be good or bad. Throughout the play, Hamlet thinks things through. It would not be out of order to say that Hamlet over-thinks things. As Hamlet proceeds to avenge his father, he thinks about things from all angles and this makes him delay any action he considers. The fact that he considers things from all angles is a very good trait for any leader to have. However, if he considers things so extensively, to the point that he's always delaying action, this could be bad in the case of whether or not to go to war, sign a treaty, help his own people with a government service, etc.
Considering that Hamlet alienates himself from many of those around him (his mother, Ophelia), as a part of his revenge plan, this would bode poorly for his role as a king. This feigned "madness" was all part of his plan, but this would not be efficacious as a king because part of being a leader is keeping many people happy: essentially, being diplomatic. Fooling everyone with a false persona might help in certain situations, but this kind of dishonesty might eventually lead to distrust and poor relations with other royal personnel and neighboring heads of state.
But, if we consider that Hamlet becomes king after killing Claudius, it is likely that Hamlet would have learned from his mistakes (alienating people and delaying action). Hypothetical evidence that he would have learned from those mistakes is shown simply by the fact of his extreme intelligence. Only an intelligent person would consider things from all angles in addition to considering the philosophical implications of one's own actions. Only an intelligent person would learn from his mistakes.
If Hamlet, upon becoming king, learns to use diplomacy (rather than feigning madness) while negotiating his own delay (over-thinking) with efficiency, he would make a very good king. I'd take an over-thinking, delaying king over a thoughtless, quick-acting king any day.
What is promising about Hamlet is his thoughtfulness (in spite of his delay). Therefore, consider a hypothetical situation in which Hamlet has to decide whether or not to go to war. Judging by his past, he would think it through and make sure that X country deserved to be attacked. Remember that his delay has to do with violence, something he considers as a last resort. Going to war as a last resort could be considered a thoughtful characteristic.
Hamlet needs to kill Claudius, but he must do it at the right time and only in a moment when Claudius' guilt is apparent to others.
The play's the thing
Wherein I'll catch the conscience of the king. (II.ii.599-600)
This kind of thinking, almost judicial, would be a good trait for a king because he would be strategic (doing things at the right place, at the right time) and he would look for opportunities to get public support for what he wants to do.