The request made by the ghost for Hamlet to kill his uncle seems a bit overboard to me, and a less radical request would be more appropriate. A more valid offer would be if the ghost wanted to help Hamlet prove to everyone how King Hamlet was killed by Claudius and let Claudius suffer the consequences of murder. If Hamlet would attempt to prove how his father died, then Claudius would eventually be put to death. By killing Claudius, Hamlet is making himself out to be the bad guy, and unless he can pull off the crazy act he will probably be killed also. The murder of King Claudius would seem to everyone as just an act of violence or madness committed by Hamlet.
I think Hamlet also feels obligated to kill when the ghost tells Hamlet he wants him to take Claudius out. How is Hamlet going to refuse a request from his dead father? Even though Hamlet hates the new king and wants revenge for his dad, knowing that you must kill someone has to stir up your emotions. After Hamlet talks to the ghost the depression Hamlet is going through is evident. He compares himself to a slave, a slut, and a coward. .
We can tell that Hamlet is going to try to kill Claudius right after his conversation with the ghost. He makes Marcellus and Horatio swear that they won't say anything about the ghost when Hamlet starts acting crazy. I believe he starts to doubt whether or not he should kill Claudius, because before Hamlet will kill Claudius he has to assure himself that Claudius was the one who killed his father. He does this by setting up a play and watching the king's reaction. .
Hamlet would not be revenging his father's death by killing Claudius. The ghost of King Hamlet is only doing more harm to Hamlet. Hamlet's father just died, he's upset with his mom for marrying so quickly and on top of all that his father just rose from the grave and told him to kill his uncle. I do believe Hamlet needs to revenge his father's death, but revenge his death by other means than murder.