Prior posters have suggested that one of the central themes was the interplay between reality and fantasy. From our modern perspective, that makes for an interesting discussion point; however, in Shakespeare's day, those fantastic elements were indeed real! Elizabethian audiences would accept the ghost of Hamlet's father as a ghost, not as we would, as a psychological construction from the machinations of Hamlet's feverish brain. The unreal was very real to these people.
An important theme is that of unraveling truth from falseness. For example, from Hamlet's perspective, is Clausius sincere in the behavior he displays before Hamlet? Is the ghost Hamlet sees his father or a demon? Is revenge his obligation of is it usurping the power of God?
To me, one of the most interesting themes in this work is simply the moral choice Hamlet faces. Should he kill Claudius, or should he leave Claudius's fate to God? Many people routinely assume that Hamlet should of course kill Claudius, and one of the big debates the play has generated involves why he waits so long to act -- why he "delays." However, a good case can be made that Elizabethans would have been quite conflicted about what Hamlet should have done -- about what was the proper moral course. Eleanor Prosser has written an intriguing book on this subject, and so has Arthur McGee.
I think one of the most important themes is the role of Fate and destiny in one's life. Hamlet comes to the ultimate conclusion in Act 5 that "there is a divinity that shapes our ends rough hew them how we will" and that ultimately, "the readiness is all." What he means is, there is a fate at work in our lives that we can only recognize and REACT to, we can't control all of the aspects of our lives. All we can do is be ready to act or react it the circumstances of our lives. This realization finally allows Hamlet to act, rather just think about acting.
I, also, like the "interplay" between what is real and what is not. I find that, in life, we struggle with this idea on a daily basis. As a teacher, I am constantly discussing with students what is real and what isn't. Therefore, that theme, for me, is one of the most important.
In accord with the previous poster, here is a link that provides an interesting discussion of the theme of pretense in Shakespeare's Hamlet:
In the preface to his published criticisms of Shakespeare's plays, the renowned critic, Harold Bloom, wrote that Shakespeare taught Freud. Certainly, Hamlet is an intriguing study of the depressive personality and its efforts to hide itself.
Whilst I obviously agree that mortality and death are major themes, I'm not convinced that they are the greatest, most interesting or significant in the play.
Personally, I would focus on the interplay between the real and the unreal; the physical and the imaginary; the actual and fantasy. Hamlet constantly ruminates and dwells on the difference between appearances and reality. We can look at his outburst too Gertrude that his grief exceeds mere appearance: he feelings are more than something that merely "Seems" to be grief. We can look at his outburst at Ophelia, complaining that God has given her one face but she paints herself another. We can look at his suspicions that the Ghost may be a counterfeit sent by the devil or a fiction conjured up by his own overwrought brain. We can look at his interview with Gertrude in her chamber, drawing her attention to the pictures and appearance of the men she has married and trying to draw out a reliable reality from them. We can look at his reaction to the actors and the comparison of his unactioned real grief and the actor's counterfeited convincing grief at Hecuba. We can look at the age old unanswerable question of Hamlet's own "antic disposition".
Throughout the play, the theme of the real and unreal overwhelm Hamlet. If we turn to the graveyard scene, the image of Yorick's skull for me is an image of the final only truth at the heart of man: once the acting and attitudes, the dissimulations and deceits, the disguises and make up, the lies and set faces are stripped away, the only honest truth in humanity lies in the stark visceral reality of our bones.
I agree completely with the previous post, and would add that Hamlet seems fascinated with the mystery of death throughout the play, whether he is contemplating suicide, remembering Yorick, reminding Claudius that a king may progress through the "guts of a beggar," or talking to the ghost of his father. Death is closely tied to the other great theme of the play, revenge, in that it is both the cause and the consequence of revenge, for both Hamlet and Laertes.
I think to answer this question we have to turn to the most famous speech that comes from this play, and arguably all of Shakespeare's works: Hamlet's famous "To be or not to be" soliloquy. The important theme that this speech points towards is the way that mortality and chance are intertwined in the lives of the characters. Note the way in which death and the meaning of life are such important topics for the confused Hamlet, and the character of the Ghost does not help clarify what it is precisely that awaits us when we die. Consider the importance of Act V scene 1 and the famous image of Hamlet handling the skull of Yorick:
Alas, poor Yorick! I knew him, Horatio: a fellow
of infinite jest, of most excellent fancy: he hath
borne me on his back a thousand times; and now,
how abhorred in my imagination it is!
Death is shown as a force that cuts down the vitality of man in its prime, as indeed it does for nearly all the central characters. Death, however, is also shown to be a result of circumstance in the case of Gertrude and Polonius. Ophelia's suicide too seems to be a result of life getting too much for her. The overwhelming theme of this brilliant play therefore is of the inevitability of death and the way that chance plays such a key role in when we come to "shuffle our mortal coil."