How is Hamlet an exploration of the human condition?
First, you have to define the human condition. And this is no easy task. Generally speaking, a condition is the state of being, appearance, quality or working order. A condition, in medical contexts, can mean a state of health or disease. The human condition deals with the state of being human. You tend to see the human condition expressed in terms of the search for meaning, facing mortality, dealing with loneliness, relationships and the limitations of freedom.
It's a very broad topic. The human condition is something we all face, so it is different for each person, even though it is supposedly a universal quality, something that pertains to all humans. Another way of putting it is, for the individual, the human condition concerns what it means to be alive and, socially, what it means to be with others. And this manifests in different ways: how we see the world, how politics and economics affects our lives, etc.
Hamlet deals with a lot of these broad philosophical themes. In fact, in the “to be or not to be” speech in Act 3 Scene 1, Hamlet directly addresses a fundamental and profound question on the human condition: is it better to live and face life's struggles or is it wiser to distance yourself or even commit suicide. Clearly, Hamlet is at his most desperate here, contemplating suicide. He is so obsessed with grief, anger, and his plan for revenge that he has lost all enjoyment in life. The state is in at this point, his condition and the condition of Denmark (“rotten”) is deplorable to be sure. But Hamlet has allowed himself to become mired in it to the point that he is quite desperate. In the end, he chooses to live at least to carry out his revenge. And despite the fact that his only reason for living is to avenge his father, he does at least face his condition rather than run (suicide) from it. That is, he decides to take the “noble” route and deal with his, and Denmark's, condition:
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or take arms against a sea of troubles,
And by opposing, end them. (III.iii.57-59).
Begrudgingly, Hamlet does choose life. At least for the sake of exposing Claudius, Hamlet decides that it is nobler to deal with unfortunate events than it is to "end them," meaning to ignore them or avoid them altogether by ending his life.
Hamlet is a dramatic exploration of the duality of the human condition. First, Shakespeare emphasizes twos all through the play. The play opens with 2 guards. The opening line is spoken twice. There are two ambassadors to Norway, two English ambassadors and two childhood friends of Hamlet. The ghost appears twice in the first scene interrupting two chronographs that precede the Ghost. Laertes leaves twice. Man and woman is one flesh. The play within the play (which is 2) has two names. There is the parallel construction of sentences, phrases and lines. There is the use of paradox, oxymoron, hendiadys and syllepsis, on and on. See Part One of Shakespeare's Language by Frank Kermode for more detail. Also see See George T.Wright, "Hendiadys and Hamlet."
What Shakespeare juxtaposes as thematic elements are various dichotomies (dualities) of life such as image and reality, love and lust, god-like and bestial, remembrance and oblivion, resolution (thought) and action, blood and judgment, nobility of war and savagery of revenge. Hamlet's youthful idealism runs into this duplicitous world of adults and the hi-jinx ensue.