He has come back to Denmark and now feels that "Denmark's a prison." The audience can see that Hamlet is not naturally a soldier and finds the act of revenge a difficult task. For example he does not kill his uncle straight away. He takes time to get enough courage to finally complete the task although other factors such as Hamlet needing proof are also influential in his decision.
Hamlet is seen as a tragic hero and the flaw in his character is his inability to take action promptly. He has to think about what would be the best thing to do first, and then take action. If Hamlet had taken action at the first opportunity and killed Claudius while he was praying, Claudius" soul would have gone to Heaven. Obviously Hamlet does not want this to happen. He also faces the problem of requiring proof that Claudius is the murderer, and refuses to take revenge until his has proof. This is why he decides to use the play within a play to provoke a reaction from Claudius. However this eventually leads to his downfall as obviously if Hamlet had taken action swiftly Claudius would not have suspected Hamlet of trying to murder him and would not have tried to have Hamlet killed. It is clear that Hamlet was heir to the throne so although Hamlet wants revenge for his father's sake, he also wishes to avenge himself.
The first time we see evidence of revenge is when the ghost of King Hamlet appears. Before the ghost appears Hamlet is obviously upset over the death of his father and his mother's marriage to his uncle. The ghost warns Hamlet "so art thou to revenge when thou shalt hear." Hamlet also suspects murder. "I doubt some foul play." He however has no proof of it until the ghost appears. "Revenge his foul and most unnatural murder." The ghost reveals that the murderer was Claudius "the serpent that did sting thy father's life now wears his crown" and although Hamlet still does not have any real proof he sees it as an "honest ghost" and vows revenge on his Uncle.