So basically I have to write a 3 paragraph paper based on this question. I would have to also answer what influences contribute to this struggle and how the theme is developed throughout the play. Also, I would have to provide textual evidence. I was wondering if anyone would like to help out :)
While the soliloquies of Hamlet are what advance the action of the play, their contents are certainly instances of where Hamlet's inner struggle between melancholy and action occur. In the first soliloquy, his original intent to avenge his father conflicts with his fear of regicide; his awareness of his mother's reprehensible behavior countered against a filial love occurs in another; his disgust for life against his fear of going to hell if he commits suicide is in yet another; and, finally, his inaction set against the fortitude of a "delicate and noble prince" is the final soliloquy, one which pushes him to action.
A study of three of the soliloquies might be an approach to your topic.
Another kind of passivity in Hamlet is Hamlet's need to continually reassure himself of his uncle's guilt. He wants to trap him with the play ("catch the conscience of the king"), he continually asks Horatio's advice, and he asks the Ghost again in Gertrude's chamber. It's likely his own commitment to morality which sets up these speedbumps, so to speak, though it certainly can look like indecision or procrastination.
I can give you a few other ideas that are somewhat related to your topic.
Characters other than Hamlet also suffer from this action/inaction dilemma. Ophelia's death is caused by her passivity. She slips into the water, and sings songs, and does not attempt to save herself.
Whether her death is a suicide or an accident is a point of humorous debate by the gravediggers. If a man goes to the water, then it is suicide, but if the water goes to the man, then it is accident, or fate.
This nebulous state is further developed between Laertes and the priest at Ophelia's funeral. The priest gives Ophelia only an abbreviate service because he suspects suicide, but Laertes wants his sister to have the full rites because he feels her death is an accident. Ophelia's death is indicative of her life. She literally drowned in a "sea of troubles," unable and perhaps unwilling to fight the current and save herself.
Gertrude is also passive. After Hamlet, in Act 3, convinces Gertrude that Claudius murdered Hamlet's father Gertrude reacts in horror: "Thou hast cleft my heart in twain." Yet, Gertrude takes no action. She feebly defends Hamlet's murder of Polonius, but continues to stand by Claudius, to the point of holding Laertes back when he attempts to rush Claudius.
We don't know Gertrude's thoughts as we know Hamlet's. But her inability or unwillingness to act on what Hamlet tells her is curious. Even in her death, Gertrude will not be decisive. She warns Hamlet about the drink, but not about the king.
Even Claudius is passive. After the play within a play, Claudius cries, My offense is rank. It smells to heaven." Even though he can analyze the various purposes of prayer, and reason about divine justice, he cannot bring himself to pray. He rises from his knees without praying, dismissing the action with "All may be well." Claudius cannot take the necessary action to truly do penance for his crime. He cannot give up his crown or his queen.
The entire subject of the "To be or not to be" soliloquy is Hamlet's examination of his failure to act. The first sentence suggests a two-fold meaning. He is perhaps speaking of suicide, but he is ultimately asking whether it is more noble to act suffer silently and not act out, or to take action against the wrongs of world even if they can't be defeated and instead they defeat him. He goes on to discuss sleep as death, but the real heart of the soliloquy comes towards the end when he questions why people don't just kill themselves, but concludes with the following: "Thus conscience makes cowards of us all, and thus the native hue of resolution is sicklied o'er with the pale cast of thought."
Hamlet realizes that human beings have all kinds of great intentions to act, but that once we think about those actions or the potential consquences of those actions, we stop because we fear the future and the unknown that the future brings. There are no guarantees in life and in so many instances, "the native hue of resolution (ACTION) is sicklied o're with the pale cast of thought" (INACTION). And enterprises of great pitch and moment (ACTION) with this regard their currents turn awry and lose the name of action" (INACTION).
Hamlet is completely aware of the fact that he has not acted on the request of the ghost in any forceful way. While he can justify the lack of action under the cause of needing to know the truth of the ghost's accusations, it is still on his mind that he hasn't done anything impressive and aggressive against his father's murderer, and this lack of action troubles him.
This could be quite a lenghty topic, so for a three paragraph paper, I would suggest choosing one instance of Hamlet's struggling to act, so that you might adequately show what influences contributed to this moment and how the moment is further developed in the play.
The moment that comes most immediately to mind, and for me the most obvious instance of procrastination, is when Hamlet decides not to murder Claudius after he believes that The Mousetrap performance has confirmed the guilt of his uncle. This is Act III, scene iii, very nearly what could be considered the mid-point of the play. The major influence that contributes to this opportunity appearing to be right for Hamlet, is the confirmation of Claudius' guilt upon witnessing the murder in the play. Yet, Hamlet finds another reason for procrastination. Claudius is kneeling in prayer. Hamlet says:
...That would be scann'd
A villain kills my father, and for that
I, his sole son, do this same villain send
Why this is hire and salary, not revenge.
Hamlet is alluding to the fact that, to revenge his father properly, he must ensure that his uncle goes straight to hell, so that he might experience the torment that his father seems to be experiencing for his misdeeds in life. I.E., his father was not able to pray for forgiveness before his death, so why should Claudius?
This moment has many repercussions throughout the rest of the play -- the murder of Polonius in Gertrude's closet, the exchange of the lives of Rosencrantz and Guildenstern for Hamlet's, Ophelia's madness and death, and, of course, the bloodbath of deaths at the end of the play. All these would have been avoided if Hamlet had acted upon his moment and killed Claudius at prayer. Of course, the irony of this moment of hesitation is that Claudius rises from his knees and says:
My words fly up, my thoughts remain below.
Words without thoughts never to heaven go.
So, Hamlet could have killed him at that moment, without, as he feared, sending Claudius to heaven.