Identify the five primary soliloquies which Hamlet delivers in the play. They can be foundin Act I, scene ii, line 129; Act II, scene ii, line 549; Act III, scene i, line 75 (the famous "to be or not to be" speech); Act III, scene iii, line 73; and Act IV, scene iv, line 33.
Understand the overall language of Shakespeare and why he wrote that way. Shakespeare wrote in early modern English, which involved a certain amount of antiquated phrases, but was essentially little different from our language today. More importantly, he wrote using poetic verse--stressed inflections and structures which attained a rhythm when spoken unlike anything ever heard before. Hamlet's soliloquies aren't intended to sound "like regular people speak." They're closer to poetry or song than normal dialogue. By examining them and seeing how richly the phrases roll off the tongue, you can understand what's so special about Shakespeare.
Analyze Hamlet's frame of mine during his soliloquies. Throughout most of the play, he's extremely depressed over the death of his father, and disturbed that his mother would remarry so quickly. Many of his soliloquies express a desire to commit suicide. Others extol the virtues of his father while condemning the failures of his uncle. If you know how Hamlet feels during his speeches, their meaning becomes much more clear.
Keep Hamlet's tragic flaw in mind. Shakespeare's great tragic figures all possess a flaw--a failure in their character--which prove their undoing. Hamlet's is indecision (or more precisely, inaction). He knows he has to kill his uncle and yet he can't bring himself to do it. That waffling crops up several times during his soliloquies. Consider his second, in Act II, when he arranges for a play to trigger the king's guilty conscience, or his fourth, in Act III, when he spies his uncle praying and declines to kill him because he's afraid the man's soul will go to heaven. He expresses a need and a desire to do the deed during these speeches, but always finds reasons, excuses and justifications to avoid going through with it.
Consider the general themes of "Hamlet." One of the overarching ideas in the play is corruption hiding behind a pleasing surface. Friends and family members cannot be trusted, and yet cover up their ugly deeds with pleasing words and glib justifications. Hamlet often rants against their behavior in his soliloquies and draws attention to their hypocrisies. For instance, in his first soliloquy, he refers to the world as "an unweeded garden": a reference to the hidden corruption around him.