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The primary reason Hamlet sets out on this quest to avenge his father's death is that the ghost of his father asks him to do so:

If thou hast nature in thee, bear it not.
Let not the royal bed of Denmark be
A couch for luxury and damnèd incest. (1.5.86–89)

And Hamlet commits himself to undertaking this role:

Haste me to know ’t, that I, with wings as swift
As meditation or the thoughts of love,
May sweep to my revenge. (1.5.34–36)

Hamlet knows that his uncle has stolen his father's wife, crown, and life, which leaves him reeling to find his own footing. After making this vow to his father, the prince spends most of the play hesitating to follow through on the vows he's made. He is utterly torn by his father's desires to exact revenge and his own inability to take his uncle's life—for reasons he can't even come to terms with himself. In act 4, he is still perplexed about why he allows Claudius to live:

I do not know
Why yet I live to say "This thing's to do,"
Sith I have cause, and will, and strength, and means?
To do't. (4.4.43–43)

In his heart, he bears loyalty to his father not only because of their relationship but also because Hamlet sees him as his true king. Thus, he feels compelled to submit to the will of his king, as any loyal subject would. Hamlet longs to be the noble son and loyal subject that his father desires.

Yet even in the end, Hamlet never fully achieves revenge. He ends up killing Claudius partly in revenge for poisoning himself and inadvertently poisoning his mother as well. He does this rashly, which stands in sharp contrast to his great hesitation in murdering Claudius as an act to avenge his father.

Although his father asks his son to avenge his death, Hamlet struggles with the weight and implications of this act to the point of spending the majority of the play avoiding it. Perhaps on some level, Hamlet rejects this philosophy of revenge and the world of corruption in which he is embedded.

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Hamlet feels the duty to revenge his father's death for several reasons. The most important one is that his father's ghost has come to visit him specifically to incite him to do so. The Ghost says:

If thou hast nature in thee, bear it not.
Let not the royal bed of Denmark be
A couch for luxury and damned incest.

This brings up a second reason why Hamlet wants to murder Claudius. Gertrude has been widowed, deceived, and corrupted by Hamlet's villainous uncle. Hamlet believes his mother is committing adultery and incest, and for some time he suspects that she might have even been an accomplice in her husband's murder. Hamlet feels that he, himself, along with his family has been besmirched and dishonored by the hateful Claudius.

Hamlet has a third reason for wanting to assassinate his uncle. Claudius not only stole the crown from Hamlet's father by murdering him, but he stole the crown from Hamlet, who should have been his father's rightful successor. Claudius took advantage of the fact that Hamlet was away at Wittenberg to get himself elected king.

In Act 5, Scene 2 of the play when Hamlet is talking to Horatio, Hamlet enumerates his reasons for needing to assassinate King Claudius:

Does it not, think thee, stand me now upon—
He that hath kill'd my king, and whored my mother;
Popp'd in between the election and my hopes;
Thrown out his angle for my proper life,
And with such cozenage—is't not perfect conscience
To quit him with this arm? And is't not to be damn'd
To let this canker of our nature come
In further evil?

In addition to the reasons Hamlet had from the time he encountered his father's ghost, Hamlet (as he tells Horatio) now has the additional reason that Claudius tried to have him killed by sending him to England with a letter calling for Hamlet's execution. If Hamlet kills the king now, he will be acting in self-defense, because Claudius will surely act publicly or secretly to have his nephew killed, especially when he learns that Hamlet forged a letter which led to the executions of Rosencrantz and Guildenstern when they reached England. Horatio responds to Hamlet's above-quoted question by saying:

It must be shortly known to him from England
What is the issue of the business there.

Once Claudius learns that Hamlet knows he intended to have him beheaded by the English, Claudius will realize that his own life must be in danger. In other words, both men will be acting in self-defense. Claudius knows that Hamlet will try to kill him to keep him from killing Hamlet. And Hamlet knows that Claudius will try to kill him to keep him from killing Claudius! The news from England has not yet arrived with the English ambassadors, so both Hamlet and Claudius can continue playing cat-and-mouse with each other before and during the fencing scene. Claudius is already dead when the English ambassadors arrive.

AMBASSADOR:The sight is dismal;
And our affairs from England come too late.
The ears are senseless that should give us hearing
To tell him his commandment is fulfill'd
That Rosencrantz and Guildenstern are dead.

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Why does Prince Hamlet seek revenge on his uncle, King Claudius, in Hamlet?

Hamlet seeks revenge because he is informed by the ghost of King Hamlet, his father, that his brother Claudius poured poison in his ear and killed him. 

Already distraught over his mother's precipitous marriage to his uncle Claudius, who has become king after the death of his father, Hamlet vows revenge on Claudius after the ghost of his father informs his son that his death was caused by Claudius, who wanted both the throne and his wife Gertrude.
In Act I, Scene 5, King Hamlet's ghost tells Hamlet that while he took his customary nap in his orchard,

With juice of cursed hebona in a vial,
And in the porches of my ears did pour
The leperous distilment, .... (1.5.60-63)

Claudius also has 

...won to his shameful lust
The will of my most seeming virtuous queen. (1.5.5-6)

After hearing these words from his father's ghost, Hamlet is incensed and vows revenge--

And thy commandment all alone shall live
Within the book and volume of my brain (1.5.102-103)

--asserting that seeking revenge will be foremost on his mind. Further, Hamlet reviles his mother for her having married Claudius: "O most pernicious woman!" (1.5.105).

Hamlet also becomes suspicious of King Claudius's motives for having killed his father because he believes that there is more involved than just lust for Gertrude. For one thing, he worries that Claudius may continue his murderous path in his desire to remain king. And yet, Hamlet is stalled by his love of words as he scrutinizes every nuance of word and thought until his resulting indecision causes him more dilemmas.

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