Hamlet’s inner struggles revolve around whether or not he should kill the king to avenge his father. He is told by the ghost of his father that he needs to kill his uncle. He is not happy with his uncle to begin with, because he did not feel that it was appropriate for him to marry his mother. Hamlet feels conflicted about the process of murdering his uncle. His struggles bring chaos to Denmark.
The approach that Hamlet takes is a little meandering. He decides to pretend to be crazy so that he has free reign. In the process, he really hurts Ophelia, who thought he loved her. His crude behavior toward her saddens and confuses her. His mother and uncle are also confused. They even ask his friends Rosencrantz and Guildenstern what is going on with Hamlet, and his uncle finally sends him away while giving them instructions to kill him.
It is not that Hamlet never has any chances to kill his uncle. Hamlet has a chance to kill Claudius when he sees him kneeling and he is vulnerable. He thinks that his uncle is praying, and doesn’t want to kill him then because he feels that he will go to Heaven instead of Hell. It turns out that he is not praying, because he found himself unable to do so, but Hamlet didn’t know that.
[Rising] My words fly up, my thoughts remain below:
Words without thoughts never to heaven go. (Act 3, Scene 3)
Hamlet's idea to have a play with a plot that is similar to the murder doesn't work out either. He seems to have thought that his uncle would just confess and that would be that. He doesn't.
Hamlet’s frustration with his mother leads to inner conflict as well. He is angry at her for marrying his uncle so soon after his father’s death. He is a frustrated, confused, and angry young man. This is why he accidentally killed Polonius. He didn’t mean to, because Polonius had not wronged him. However, that did not stop him from using Polonius’s body to continue his madness act.