Well, Hamlet DOES go after the ghost to find out what exactly is going on. Hamlet DOES create a plan to feign madness. Hamlet DOES confront his own mother with her own villainy. Hamlet DOES kill Polonius in an attempt at action. Hamlet DOES hatch a plan with the players to perform The Murder of Gonzago. Hamlet DOES confront his girlfriend, Ophelia. Hamlet DOES arrange for Rozencrantz and Guildenstern to be killed instead of Hamlet himself. Hamlet DOES Hamlet DOES participate in the match at the end of the play. (The irony is, I believe Hamlet's tragic flaw is, in fact, inaction. It just goes to show that you can "prove" opposing points of view by using the text. Ah, the shock value involved in teaching literature!)
There are many of the opinion that Hamlet procrastinated. However, we might consider that Hamlet's "inaction" is his action. Do not forget that Hamlet was a thinker. At the beginning of the play we learn he had been away at school before his father's death. He wants to return to his studies, but both Gertrude and Claudius persuade him to stay on for his mother's sake. It is no surprise that when he is confronted by king Hamlet's ghost that he must ponder and think and question before taking action. That is what students do.
In all fairness, Hamlet is a doer as opposed to a procrastinator. He suspects Claudius has murdered his father, but he wants to be certain.
Hamlet has the actors reenact the murderous scene, searching for signs of guilt from Claudius.
Hamlet is trying to find the truth. He wants to be absolutely certain that Claudius has indeed committed the murder.
There is something to be said for Hamlet's intentions. He needs proof. When someone accuses another of murder, this is a serious accusation. This is something that should not be taken lightly.
Although Hamlet is trying to do the right thing, he has waited too late. Claudius suspects that Hamlet knows. He arranges a fight between Laertes and Hamlet.
By the end of the fight, Gertrude, his mother, has drunk the poison, Laertes has injured Hamlet and ultimately Hamlet, finally, has his revenge on Claudius, thus killing him in the end, right before Hamlet dies.
In the ROMEO AND JULIET group, as I understand some of the postings, Romeo should have thought things over more and the Friar should have consulted other authorities. In HAMLET, we have more questions yet. The "To be, or not" monologue can be interesting to memorize and then recall during a tedious afternoon. The evidence is such that it is more likely than not that Hamlet is aware that his uncle is present when he says "To be" and is speaking to the King. One might then argue that Hamlet is not as sociable as he should be, though he is experiencing grief.