Hamlet has promised the ghost of his father to avenge his death. Upon deliberation, Hamlet finds himself in a quandary: If he kills Claudius, he can be severely punished for regicide, especially if he cannot prove that Claudius has murdered his father. In addition, Hamlet wonders how he will help Denmark if he is convicted. Who will then be king?
It is this tortured self-debate that consumes Hamlet, causing him to delay action. Melancholic by nature, poor Hamlet wonders if he should just commit suicide--"To be, or not to be..." but his "conscience doth make a coward" of himself as he knows that suicide is a mortal sin and he will condemn himself to eternal punishment. This soliloquy of Hamlet's and all his others, which are what propel the action of Shakespeare's play, are all connected to Hamlet's sense of obligation to his father to avenge his death.
The play is essentially about what is eating at Hamlet. From the beginning, when he sees his father's ghost, Hamlet does not know what to do. He thinks his father wants him to get revenge, but he's not sure if that really was his father or his mind playing trciks on him. Whether or not it's all in his head or not, and what to do about it, does eat at him.
Um, ... yeah it is. This strikes me as a quiz question that a teacher might ask in order to find out if a student "really" read the play. There is absolutely no doubt that vengeance is eating at Hamlet. His dad comes back as a ghost and asks Hamlet to avenge the murder for goodness sake! Of course Hamlet is going to try his best. Then our Hamlet thinks and thinks and thinks, ... and tries to kill Claudius but stops because he's praying, ... and Hamlet thinks some more, ... and then wonders more about the ghost and so tries to catch Claudius in the sin through the play-within-a-play, ... and Hamlet then thinks so much that he almost doesn't want to live himself. It goes on and on. Hamlet goes so far as to act (act?) crazy in order to catch Claudius in the act. All the while, swirling around in Hamlet's mind is the idea that his dad is dead because of this guy! Eating at Hamlet? ... I think SO!
I don't think that it is the idea of vengeance that is eating at Hamlet. Hamlet knows that he should take vengeance. He knows that Claudius's actions deserve punishment. The problem is within his own personality. Instead of taking action immediately, he wants to think about the meaning of life and about whether life is worth living, things like that.
I don't think he is tied up in knots worrying about vengeance. I think he's worrying about life in general.
Hamlet's whole dilemma concerns avenging his father's murder. Historically and culturally, it is Hamlet's duty to take the life of the man (Claudius) who took his father's life. In a non-christian world this would be easy: however, Hamlet lives in a Roman Catholic world such an action is a mortal sin.
For Hamlet to achieve his goal, the death of Claudius, and not loose his soul, is his dilemma. He has seen his fahter's ghost and the suffering he is enduring since he died with sin on his head and he must "work" it off in purgatory. He wants Claudius to suffer in hell for his sins but in the process, he does not want to join his uncle.
Before he even desides on how to do it, he must prove to himself his uncle's guilt. He does not want to been trapped by satan. He uses the mousetrap to trap his uncle.
When he comes across Claudius praying, he has a chance but beause the man is in prayer, a state of grace as it were, he doesn't kill him since it would send him the heaven. Of course what Hamlet doesn't know is that because of his guilt, Claudius cannot pray.
Avenging his father's murder blinds Hamlet to everything else and a number of people die as a result. Vengence is at the heart of the play but also there is the question of the result of it. Do flights of angels sing him to his rest as Horatio says at his death?