Hamlet is too sensitive to be a good king. He gets all tied up in knots, talks to his dead father's ghost, considers committing suicide and also commits murder himself. He seems to feel everything deeply. Perhaps as he got older he would become more hardened?
Hamlet is far too indecisive to be a ruler. Deliberation is wise, but there are times when action must be swift or kingdoms can be lost. For instance, Hamlet's indetermination is what has contributed to the demise of Ophelia.
Hamlet lacks the confidence to be a successful leader. In each of the soliloquys in the play he is moaning about some aspect of his situation and questions whether he has what it takes to get the job of vengeance done. He calls himself a ""john-a-dreams," "a rogue and peasant slave" and "a coward." He recognizes the confidence of Fortinbras in Act 4 as he marches his troops towards Poland to fight over useless land. He sees that even Horatio has a nicely "commeddled" balance of "blood and judgement." He whines about his circumstances saying "o cursed spite that ever I was born to set it right" and "how all occasions do inform against me." Without some confidence, he would absolutely fail.
One of the reasons why Hamlet would make a terrible king is because he does not act. His constant vacillations between what he should do and what he doesn't do would foment revolution in an instant. If regal leadership is defined by a sense of leading people, inspiring them to action, and ensuring that a course of government is laid out, Hamlet is not your man. Hamlet features a sensibility whereby "function is smothered by surmise." This is not what makes a successful leader. There is a parade of historical figures who have failed at leadership at their inability to take action and the constant waffling in decision making abilities. For this, Hamlet would not make a very good leader. At the same time, I think that Hamlet's lack of control of his own emotional state would not be suitable for political leadership. Hamlet is seeking to fully grasp his own emotional sense of identity and in wrestling with this, I think that failures on a political level would be evident. Again, there are numerous examples of poor political leadership on the part of individuals who could not sort out their own emotional challenges in a clear and focused manner. Hamlet would belong to this group. His inability to be emotionally honest with Ophelia is one such example where his lack of clarity in his own emotional schematic results in poor decision making. This is not reflective of someone who would be successful in political office.