How Does Hamlet Kill His Uncle

Yes of course this is the last problem, in Hamlet Hamlet has to kill his own uncle, on a hunch that he killed his father. Now yes, Hamlets father did come to Hamlet, to say “The serpent that did sting thy father’s life Now wears the crown.” (I,v,44-45) but, can anyone truly believe Hamlet when he says, that his dead father came back and told him? What makes this all worse is the fact of, as Charles Alexander said, “You aren’t crazy... unless you don’t realize you are crazy.” This problem is more of a up front problem, he can't kill someone, it's against the law. Even in the older times you would be punished to the full extent of the law. He also can't kill his uncle, that's bad form all around. People saw killing your family as something even