The only character in Shakespeare's Hamlet that is even slightly like a character in A Midsummer Night's Dream (in my mind) is Claudius.
This is a very difficult comparison, First, Hamlet is a tragedy; A Midsummer Night's Dream is a comedy. Also, some of the similarities are based upon each man's role in his respective play. Theseus is the Duke of Athens and commands a great deal of power—he upholds the law and people come to him with their grievances, on which he passes judgment. For example, when Egeus' daughter Hermia refuses to marry Demetrius (even though this is a comedy), Egeus' implores Theseus to force Hermia into the marriage: if she refuses, she will die or enter a convent. As the leader of his people who knows the law, Theseus makes these kinds of decisions; here he warns Hermia that if she does not marry as her father has dictated, she will be put to death or join a convent and be a nun for the rest of her life, living forever without being in the company of a man:
Either to die the death, or abjure
For ever the society of men. (I.i.67-68)
Theseus is a force to be reckoned with, but he is also wooing Hippolyta, the leader of the Amazons (a warrior race of women), who he has defeated in war—this woman he has arranged to marry. We can see throughout the play that Theseus genuinely cares about what his future-wife thinks. When the lovers reappear at the end of the play, Theseus is amenable to what Hippolyta has to say about the situation.
Claudius is also the sovereign of his country, Denmark. His word is law. When he makes a request, it is followed without question. For example, after Hamlet has killed Polonius by mistake, Claudius orders Rosencrantz and Guildenstern to take him to England. (His plan is to have his stepson executed there.) Hamlet's friends don't have much in the way of integrity or loyalty to their school friend—Hamlet—but even so, they have little choice but to do what the King asks. When Ophelia's father (Polonius) and Claudius require Ophelia to spy on Hamlet, she also has no choice.
Claudius is also in a new relationship—recently married to Gertrude. It would seem he tries to please her by attempting to bring Hamlet out of his depression over his father's death.
‘tis sweet and commendable in your nature, Hamlet,
To give these mourning duties to your father;
But you must know, your father lost a father;
That father lost, lost his, and the survivor bound
In filial obligation for some term
To do obsequious sorrow. (I.ii.90-95)
That both men are leaders of their particular domains is something the two men have in common. That they are both emotionally involved in the lives of women whom they choose to please is also similar. However, while Theseus seems to genuinely care for others, Claudius proves (by the play's end) that he cares for no one but himself.