What are some major literary devices used throughout Hamlet?
Shakespeare, the Bard, regularly uses double entendres (word games), metaphors (even extended ones) and allusions. These are things that Shakespeare uses a great deal in his plays.
Double entendres: When Claudius demands to know where Polonius' body is hidden, Hamlet explains that he is at supper. When questioned, Hamlet explains that it is not about what Polonius eats, but what eats Polonius. This is words with double meanings. Hamlet does this quite a bit with the innocent Ophelia. Because he believes she has betrayed him and he cannot trust her, he torments her with insults, like "Get thee to a nunnery," and with feigned madness, like running in to her chamber half undressed. Then, after this cruel treatment, when Polonius dies, she loses her mind.
Another example of word play and double meaning is that when Hamlet arranges to have the players present a play to "catch the conscience of a King," he tells the members of the court that the play is entitled Mousetrap—what a clever way to infer that he is searching out a "rat" or some other vermin: namely Claudius.
One implied metaphor is when Hamlet explains that he has little value for his miserably saddened life, thus little fear of losing it from an encounter with a ghost. When Horatio worries about his friend approaching Old Hamlet's ghost, Hamlet notes:
Why, what should be the fear?
I do not set my life in a pin's fee; (I.iv.67-68).
What Hamlet means is that the cost of a pin (an insignificant cost as pins are usually sold many at a time for one price) is worth more than Hamlet's life, or so he feels in his tragic circumstances of a dead King and father and an abominably remarried Queen and mother.
As an allusion, Polonius notes that he once acted the part of Julius Caesar and was stabbed by Brutus. This is an historical allusion to the Roman Emperor, Julius Caesar, and one of his assassins. The allusion also provides foreshadowing of what will come because it will be the way that Polonius dies: Hiding in Gertrude's room behind a curtain ("arras"), Hamlet stabs him believing him to be a traitor in Queen Gertrude's room.
I did enact Julius Caesar; I was killed i' the Capitol;
Brutus killed me. (III.ii.98-99)
In addition to the devices listed above, you could also consider such elements of the play itself such as structure, methods of characterization, and theme. You could also look for examples of irony and motif, like the irony of death by killing coming to almost all when Hamlet was trying to avoid killing and the motif of betrayal of friends by friends.
The play is a typical five-act structure with Act 1 being primarily for exposition; Act 2 is the rising action with further complication; Act 3 brings the climax (point of no return); Act 4 is the falling action; and Act 5 is the resolution of the conflicts.
It is always interesting to consider which exact action or choice by the main character marks the climax of the play. In this case is it Hamlet's decision not to kill Claudius when he has a chance or is it the killing of Polonius?
When you consider methods of characterization you think about how the author uses a blend of what characters say, do and think (direct characterization) as well as what others say about them (indirect characterization) in order to develop the complete character. This is certainly a valuable study for many of the characters in the play.
There are many themes in the play: life/death; what happens after death; fate/providence; love/marriage; revenge; kingship; friendship; action vs. inaction. This names but a few. What others might there be?
Shakespeare used a wide variety of literary devices in all of his plays, and a lengthy essay would be needed to discuss them all. I can explain a few of them, as well as provide a link that will help you further understand them. Each example below is from Hamlet's famous speech in Act III, scene 1 ("To be or not to be"):
antithesis: contrary ideas expressed in a balanced sentence
Example: "To be, or not to be, that is the question:" (opposite ideas of being or not being, living or dying, are expressed in this sentence).
alliteration: repetition of an initial consonant sound in two or more words of a phrase, line or sentence.
Example: "With a bare bodkin?" (the "b" sound beginning bare and bodkin).
allusion: the mention of a person, event or condition thought to be familiar (but sometimes actually obscure or unknown) to the reader.
Example: "The fair Ophelia.--Nymph, in thy orisons/Be all my sins remembered" (a nymph is a mythological entity, or mythological allusion).
soliloquy: a speech delivered by a character in a play while talking to themselves but "overheard" by the audience.
Example: Hamlet's entire "To be or not to be" speech is a soliloquy; it is to himself and meant to be heard by the audience.
This is an extremely broad question. You could discuss the importance of language, especially Shakespeare's iambic pentameter. You could also focus on the language of puns (play on words), litotes, irony, repetition and alliteration. You could discuss symbols like Yorick's skull and the ghost. You could discuss day and night and how the shift in time plays an important role in understanding what is in fact "rotten in the state of Denmark."