[Second post to above answer.]
Another view emphasizes Hamlet's great qualities of intellect, philosophical leanings, strength of will and intensity of feeling. These lead Hamlet to lament the philosophical point of Denmark's loss of moral integrity thus immobilizing his will to act in a way that will contribute more to the loss of moral integrity. This view is difficult to reconcile with his behavior toward Rosencrantz and Guildenstern and, moreover, toward Ophelia. Then there is the Freudian Oedipal theory in which Hamlet will not act against the King because he psychologically and subconsciously shares the King's guilt as it was his own secret longing to murder his father and absorb his mother's love for himself.
That Hamlet's qualities are agreed upon and that he delays and then refuses to act because of some moral dilemma seems to run through all criticisms. The approach that seems to shed most light is set against the backdrop of the Renaissance and Protestantism, with his moral outrage against his mother being psychologically displaced to Ophelia making her fair game for any mistreatment be it overtly cruel or underhandedly manipulative as during the play scene.
[This answer is too long for one post, so is in two posts.]
The nature of the character of Hamlet in Shakespeare's tragedy Hamlet is a complex question that has engendered much debate over the centuries. There is even the minority opinion that the character of Hamlet does not and cannot make any sense because Shakespeare wrote his character so badly that there is no unity at all and no motive at all to be found for Hamlet's behaviors. This argument is countered with the reminder of the era of the European Renaissance during which ideas were being overturned in many respects, not the least of which was the Protestant Reformation begun by Martin Luther. It is pointed out that Hamlet was in Denmark, a kingdom that had embraced the new Protestantism, and that he was educated at the university in Wittenberg, which was Luther's university.
In keeping with the discussion about the Reformation, there are indications that the ghost has risen from Purgatory, a Catholic religion construct rejected by Protestants. Thus Hamlet's worry as to whether the ghost is a "goblin." Protestant theology asserts that souls go either to separation for God or to union with God, with nowhere in between. If a soul has gone to union with God, then what need to come back on a mission of revenge?
Critics agree on some major questions related to the character of Hamlet: (1) Why does Hamlet initially delay in taking revenge as instructed to do by the ghost? It is pointed out that many months go by and not just a brief time. (2) Why, when Hamlet relinquishes his delay in Act IV, scene iv, does he not precipitously act? There are several answers to these questions.
One is that Hamlet has to be convinced of the veracity of the ghost's message, after all, it is a little foolhardy if not plainly irrational to run out and murder a king on the word of an aggrieved ghost. Then, once convinced, or at least resigned, Hamlet is so deeply depressed that he has to reconcile himself even further to his ultimate fate. It is suggested that he lacks emotional strength to pursue his commission of revenge even though he is outraged against his mother and the new king. His depression and angst are demonstrated in his treatment of Ophelia, which by any standard is atrocious and without cause. In this criticism, it is generally agreed that Hamlet's tragic flaw is his deeply emotional nature and depression which clouds his vision and essentially paralyze him.
[Second post of answer below. I hope...]
Hamlet is a young man who is worshipped by the people for his heroics. He is also a soul torn by decisions that he has to make on behalf on his father. His father was murdered by Hamlet's uncle who then marries his mother. Hamlet is also believed to have an attraction to his mother that goes beyond mother/son feelings. He is manic and depressed. He is an ambiguous character who talks to himself often in monolog. He is a cruel lover who treats the woman that loves him badly. Hamlet is a doomed soul.