Hamlet's major flaw is also his major strength: it is his intellect. Hamlet is a modern man trapped in a situation that is decidedly Old World. Remember, at the beginning of the play, Hamlet is not present. He is away, attending university in Wittenberg, Germany. This is significant and not chosen randomly by Shakespeare. Wittenberg was the seat of the Protestant Reformation, a time of vast change. Denmark was still Catholic, and the old ways, including superstitions, were still rife across the land. Hamlet has been indoctrinated into more humanistic and scientific ways of thinking. Therefore, when he returns for his father's funeral, and Horatio reports seeing a ghost that resembles his deceased father, Hamlet is skeptical. It will take seeing the Ghost with his own eyes before he will take action.
The "action problem" is one that has been debated and discussed by scholars for centuries. Hamlet must always think things through. So thorough is he in his intellectual contemplation that he misses the key opportunity to avenge his father's death (as the Ghost has ordered him to do) by killing Claudius when the rogue king is at prayer. But in, 3.3.1, Hamlet stops himself:
Now might I do it pat, now he is praying;
And now I'll do't. And so he goes to heaven;
And so am I revenged. That would be scann'd:
A villain kills my father; and for that,
I, his sole son, do this same villain send
Hamlet has hesitated. He stays his sword as he believes that Claudius has confessed his sins, and should he kill the man confessed, he would go to Heaven, not to Hell, as both Hamlet and his dead father desire. The irony is that Claudius has not confessed. Had Hamlet come upon the man unawares just a few seconds sooner, he would have heard Claudius admit that he is not prepared to ask for forgiveness, because he is not ready to give up the spoils of the kingdom: his riches, his throne, his wife. Although he is on his knees, he cannot bring himself to get right with God.
Had Hamlet not hesitated, the tragedy to come would not have unfolded. Polonious would have lived, Ophelia would not have gone mad and killed herself, Laertes would not have been manipulated by Claudius, causing the deaths of Gertrude, himself and Hamlet.
Although the examples I have given you here pertain to the downside of Hamlet's intellect, it should definitely be noted that his intellect is also the cause of ferreting out Claudius's deceit and his mother's complicity. Consider the intricate way Hamlet "catches" the corrupt couple in his play "The Mousetrap." Also, consider the crafty way Hamlet is able to save his own skin by outwitting the admittedly dimwitted Rosencratz and Guildenstern. Additionally, Hamlet's intellect is sparkling in the scene where he plays with the clueless Polonious, running intellectual circles around the "fishmonger." It is also delightful to watch Hamlet's intellect at work when he taunts Claudius to find Polonious' body in Act 4, Scene 2:
Not where he eats, but where he is eaten: a certain convocation of politic worms are e'en at him. Your worm is your only emperor for diet:
Here, Hamlet engages in puzzles and wordplay, clever taunts and threats. His intellect is always on, he does not know how to turn it off and simply be a man of action, like his father. Again, it is both his blessing and his curse.