One of the main difficulties that Hamlet faces is that he will probably not have the opportunity to become king. Because Claudius has taken the throne by the dubious, if not actually illegal, method of marrying Hamlet’s mother, the young prince would have to wait a long time. Should Gertrude have another son, who would be the current king’s heir, he might be in line for the throne.

Hamlet must spend quite a lot of his time working through moral dilemmas. One positive feature in his character is that he wants to do the right thing. He does not want to accuse his uncle or his mother of murder unless he is sure that either or both of them is guilty. This adherence to moral values would be a very good quality in a king. However, Hamlet is also indecisive. He is more of an intellectual than a politician or a diplomat, and he sometimes fails to think through the consequences of his actions. This character flaw is evident in his dealings with Ophelia, whom he denounces as disloyal before giving her a chance to explain her behavior.

Claudius is a politician through and through. Because he loves power, he has been, and continues to be, willing to do whatever it takes to be the king. Although he might have had some misgivings before committing the deed, at the decisive moment, he killed his own brother. He seems to be a sociopath with no real affection for anyone. His marriage to Gertrude stems not from love for her but only his insatiable desire for power. Claudius is poorly equipped to govern a state, and is a far worse king than even the bookish, hesitant Hamlet would ever be.

In Hamlet, Claudius is an evil king. He murders his brother and marries his brother's wife. Claudius will pay any price to inherit the throne. He is dangerous. Not only does he kill his brother, but he creates a plan to have young Hamlet killed. Clearly, Claudius does not even care about the fact that young Hamlet is the son of his wife Gertrude. Claudius has no morals and values. He is selfish and he will ultimately destroy those around him. 

While Claudius is a despicable king, Hamlet would definitely make a good king. He is very careful in making serious decisions. While Hamlet desires to avenge his father's death, he wants to be certain that Claudius is indeed guilty. Hamlet does not make rash decisions. He is creative and intelligent in his plan to test Claudius' guilt. In the end, Hamlet is a man who deals with a "hero’s struggle with two opposing forces: moral integrity and the need to avenge his father’s murder."

Hamlet proved his integrity by not killing Claudius immediately. He waited until he was convinced of Claudius' guilt. He did not act impulsively. He has patience which is a virtue. 

Truly, Hamlet would make a good king. Of course he lacked maturity, but in time he would have matured into a good king. He has every right to be upset with his Uncle Claudius and his mother. After all, Claudius has murdered his father and his mother has rushed into a marriage with her husband's brother. This is unacceptable behavior on the part of Claudius and Gertrude. No doubt, Hamlet was severely wounded emotionally by the the events that happened at the hand of Claudius. 

No doubt, Hamlet would make a good king. Toward the end of the play, he is receptive of Laertes' apology. He receives his apology at the end of the play with good will. In his dying breath, he forgives Laertes. Even though Laertes was trying to kill Hamlet, before he dies, Laertes admits that Claudius deserved death. Laertes asks for fogiveness while claiming that Hamlet was most noble:

[Claudius] is justly served,
It is a poison tempered by himself.
Exchange forgiveness with me, noble Hamlet.
You are not guilty of my and my father's deaths.
And I am not guilty of yours.

Likewise, in his dying breath, Hamlet is concerned about the kingdom. Hamlet feels that the kingdom will be in good hands when Fortinbras takes over the kingdom.

While Hamlet is dying, he is making sure the kingdom will be in good hands. He gives his dying vote to Fortinbras:

He has my dying vote,
Tell him so, with the news, more and less,
Which has been asked for.

No doubt, Hamlet would have made a good king. Even Fortinbras claims that Hamlet would have been most royal as king. Fortinbras  gives him a respected soldier's burial:

Bear Hamlet like a soldier to the platform,
For he was likely, had he been King,
To have proved most royal, and, for his passage,
The soldiers' music and the rites of war will
Speak loudly for him.