Hamlet discovers that his father has been murdered by Claudius early in the play, in Act I, Scene 5. He is beseeched by his father in this scene to strike against his uncle:
If thou hast nature in thee, bear it not.
Let not the royal bed of Denmark be
A couch for luxury and damned incest.
Hamlet determines that he will avenge his father's death and set things right in Denmark. But he constantly puts off the act, delaying the realization of justice. Primarily he wants to determine whether his uncle is actually guilty of murder, as his father's ghost asserts. After the play, when Claudius's response convinces Hamlet beyond a shadow of a doubt that he is culpable for the murder, he still hesitates to bring about justice, deciding after happening upon the king in prayer that he will kill him when he is engaged in a less pious activity:
When he is drunk asleep; or in his rage;
Or in the incestuous pleasure of his bed;
At game, a-swearing, or about some act
That has no relish of salvation in't
Then trip him, that his heels may kick at heaven,
And that his soul may be as damn'd and black
As hell, whereto it goes.
So Hamlet delays his revenge for different reasons (his distraction with Ophelia also sets him off) and thus is only able to avenge his father's death at the climactic duel, and then only at the cost of his own life.
There are also references which contribute to the theme of delay in an ironic way. Take, for example, Horatio's claim to Hamlet that the reason for his visit to Elsinore is to attend the funeral of the former King of Denmark. Hamlet supposes that the real reson was to attend the more recent wedding of his mother to the late king's brother.
Hamlet's sardonic comment: "Thrift, thrift Horatio. The funeral bak'd meats did coldly furnish forth the marriage tables" concisely illustrates Hamlet's dismay at the seemingly hasty marriage of his mother to his uncle, Claudius. This haste contrasts sharply with Hamlet's own inaction following the revelation by the ghost of his late father who claims: "The serpent that did sting thy father's life / Now wears his crown."