What makes Prince Hamlet a unique character?

Hamlet is unique, paradoxically, because he also seems to be the universal character. He never seems irrelevant to the times or to the vast majority of attentive audience members. His play has been performed everywhere and for hundreds of years and is utterly resonant. Just as every character in the play seems to think they have him figured out, so too does each audience member sense that Hamlet speaks uniquely to his or her fundamental issues: "the time is out of joint," "I know not seems," "to be, or not to be," "I loved Ophelia."
One further quality that makes him unique is his dynamic introspection. In his soliloquies, we see a mind at work, inventing and enlarging itself as it goes. The Hamlet of Act 5 is not the same as the Hamlet of Act 1, and the work of thinking done in the soliloquies accounts for that change. When we compare Hamlet to like characters on the Elizabethan stage, this quality seems even more striking. As a revenge character, he is unique in that he both feels a passion for justice but can also think about it and weigh multiple options simultaneously. People over the years have identified in these soliloquies a modern consciousness, aware not only of an external world but of an internal one.
Hamlet is also unique in ability to embody the Renaissance Man trope. We see this especially in his relationships, where he makes himself seem naturally an equal to everyone, even while the audience recognizes his superiority. He can argue philosophy and politics, write poetry and scenes for a play, fence admirably, and engage in incredibly witty wordplay. Others seek this ideal, but Hamlet possesses it with the effortlessness (or sprezzatura) that others only affect.

Hamlet is vastly different than the other characters in Shakespeare's play of the same name, largely due to his sensitive, tormented, tragic nature. He is a character who is constantly questioning the people and circumstances around him, and who is singularly obsessed with avenging his father. He doesn't seem to express normal human interests--eating, sleeping, seeking entertainment, etc.--and eschews the companionship of others. 
Hamlet is an extremely interior character, one who we only really get to know and understand through the frequent use of soliloquy throughout the text. Through this narrative style, which almost feels confessional, we learn of Hamlet's intense internal crisis: his self-doubt, constant analyzing, and wildly shifting moods. Although not universally loved by audiences who may find him frustrating, indecisive, and inept, he is easily one of the most complicated characters in the Western canon, which perhaps makes playing him such a sought-after pursuit for many actors.