This is a really good question! There are definitely dramatic elements about the soliloquies of Hamlet, some of the most famous language in all of Shakespeare's plays, that reveal aspects of Hamlet's character that his other text does not.
First, all the soliloquies are in verse. This may seem obvious (We most often pay attention to the lines of verse in Shakespeare.), but, in actuality, the balance of Hamlet's text, at least through Act IV (when Hamlet is shipped to England), is in prose.
For Shakespeare, the choice to use verse or prose revealed a huge amount about the situation and state of mind of a character. So, Hamlet's use of verse as he converses alone onstage with audience is meant, first of all, to show how beautifully lucid and deeply thoughtful Hamlet is about his life and situation. This is meant to contrast with his putting on of an "antic disposition" and use of prose.
The use of verse in the soliloquies creates a sense of collusion between Hamlet and the audience. They share his secret (that he "is mad in craft"). Hamlet also uses verse with nearly every exchange with Horatio (who is also in on his secret) and with his mother in the famous closet scene, to whom he also reveals that his "madness" is a ruse.
Along with revealing the beautiful lucidity of Hamlet's actual state of mind, the soliloquies reveal to the audience how deeply affected he is by the state of affairs in his world. He speaks often of death, wondering what it would be like to die, to just end it all. This, again, gives the audience insight that is not present as he endlessly jokes at the other characters' (especially Polonius, Rosencrantz and Guildenstern) expense. This "antic" behaviour can truly be seen, because of the depth of feeling in the soliloquies, as a mask that covers Hamlet's pain.
And finally, the soliloquies add to the picture we have of the Hamlet who is indecisive, who cannot seem to simply act upon the revenge which is his given task. This is made even more obvious when Hamlet returns to the play in Act V. From this re-entrance, he has not another solilouqy in the play. I would say that this signals his newfound peace of mind, his acceptance of his situation. He is no longer in conflict about who he is and what to do, and, when the moment finally presents itself during the final scene of the play, he does not hesitate, but kills Claudius instantly.
It is in Act V, scene ii that Hamlet shares his new perspective on life, and this time he speaks in simple prose:
We defy augury.. . .If it be now, 'tis not to come; if it be not to come, it will be now; if it be not now, yet it will come. The readiness is all.
The "it" that he speaks of can be assumed to be the great topic of all the pain and angst of his previous soliloquies -- death. Yet now, he is quite resigned, almost Zen-like, about death. He has resolved the burning internal struggles that gave birth to the earlier soliloquies and we can assume that he dies a man at peace with himself.
There are, however, many points of view on Hamlet and his soliloquies. Please follow the links below for more.