Just to give you a look at one specific speech that demonstrates Hamlet's thinking and suggests the possibility that his thinking contributes to his delay, I'll briefly add to the exceptional answer above.
In Hamlet's famous "To be or not to be" speech (Act 3.1.56-87), he ponders the state of existence. To be is to exist. Hamlet wonders if existence, with its "slings and arrows of outrageous fortune," is worth the trouble.
He thinks about the perils of existence compared to the possible perils of the afterlife. He concludes that if one did not have to be afraid of what might come after death, one would be smarter to not exist at all. But, since the afterlife is an "undiscovered country" from which "No traveller returns," it is better to bear the "ills" we know, than to take a chance on those we don't.
The details of his speech present Hamlet's capacity for thinking, but for your purposes you need not even go into the details. What matters in answering your question is that Hamlet is thinking about the general issue of existence here, rather than thinking about the specific issue of his revenge.
If you want to make the argument that Hamlet's thinking gets in the way of his revenge, this speech is a possible piece of evidence. Hamlet can't just seek revenge, one could argue, he must first consider every angle. One could say that Hamlet should be planning revenge, here, not contemplating existence. You may or may not be correct if making this argument, but if this is your assignment this speech may help.
In Acts I - IV, Hamlet calls himself an "ass," a "coward," a "beggar," "mad north-north-west," "pigeon-liver'd," "tame," "not splenitive and rash." In short, he is anything but vengeful.
As a scholar Hamlet wants to confirm, observe, and validate motivations before giving action to passion. As a Christian, Hamlet suffers from a spiritual crisis: is Hamlet prepared to murder and, therefore, go to hell? Is he an active agent who exhibits free will or is he just following supernatural orders toward a cruel fate? How can he send his father and himself to heaven and Claudius to hell?
I'll have these players
Play something like the murder of my father
Before mine uncle: I'll observe his looks;
I'll tent him to the quick: if he but blench,
I know my course. The spirit that I have seen
May be the devil:
As an artist, Hamlet relishes artistic revenge more than actual revenge. He is more giddy getting a rise out of Claudius during the staging of "The Mousetrap" than at any other time. He prefers "catching the conscience of the king" than shedding blood.
After Claudius' guilt is confirmed by his reaction to the play-within-the play, Hamlet refuses to kill him during prayer because he fears his soul will go to heaven:
Hamlet finally accepts his new role as avenger, or at least as a man of action. At long last, he says: