Hamlet is very existential.
With few exceptions, an existentialist asserts that an individual is determined from within, not from outside forces. On the human and literary level what makes existentialists and their heroes homogeneous is their "perfervid individualism." Existentialism posits that existence precedes essence (actus essendi). Essence refers to what man is (his nature); existence denotes that "he is." Thus, "I am a man" is first part existence ("I am") and second part essence ("a man."). This is in opposition to traditional Platonic essentialism (think "Parable of the Cave"), which embraces man's essence prior to his existence. Essentialism affirms that man has an eternal, unchangeable human nature.
So Hamlet shows that Hamlet is a perfervid individual who rejects what all forms of determinism: political, familial, spiritual, and romantic. Hamlet has no universal human nature: he's not a son, or a student, or a Prince, or lover, or a coward, or sane, or crazy, or "to be," or "not to be." As an existentialist, Hamlet can be whatever he chooses to be: he can move from being a Christian to a pagan avenger, from sane to crazy and back to sane again. His family, however, tries to label and determine him: Claudius says his grief is unmanly; Polonius thinks he's crazy; his mother thinks he's a misogynist. Much of Hamlet's angst comes from begin classified as such.
This transormative process is agonizing for Hamlet, and it causes him much distress and delay. Hamlet's crisis is deciding if he is simply fulfilling a pre-determined role given him by his father ("Avenge me") or if he is an active agent in the world, committing revenge as a personal and moral choice to ride the kingdom of corruption.
In the end, Hamlet says, "Let be" ("existence precedes essence"). One must exist before one is defined:
There is excellent information to answer your question right here on eNotes, on the "themes" link below for this play. Hamlet is such a deep play and as explained on the eNotes study guide, it is much more than just an Elizabethan "revenge" play.
Although Hamlet is instructed by his father's ghost to exact revenge on his uncle, Claudius, who has murdered Hamlet's father (the former king) and married Hamlet's mother, Hamlet cannot seem to carry out the deed. Throughout the play, he constantly struggles with this. Killing a king would be an act so odious that it would affect Hamlet at his core, even though the king deserves it. The fact that Claudius killed his brother has caused "something rotten" to permeate all of the state of Denmark, so this is part of why Hamlet cannot act. Somewhere within him, he he senses that although justified, it would be wrong.
The deeper reason has to do with the prevailing religious belief at the time that vengeance was God's to dispense. When humans take vengeance into their own hands, there are dire consequences. There are many Christian beliefs and attitudes in Shakespeare's plays. Modern readers tend to forget about this.