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.