The anti-father
Instead of having a father to help him, Hamlet has an invasive uncle who has enjoyed the king's death by raising himself to a powerful position by marrying Gertrude, Hamlet's mother. The ironic inversion of Hamlet's desires shows the distance he must cross to fill the gap left by his father's death. He must correct the course of Denmark somehow, but he is overwhelmed and without the moral support of his father. Claudius's advice is horrible and offensive instead of being familial and supportive.
Laertes, the anti-hero
Although Hamlet desperately wants to go back to school for the semester, his uncle refuses to let him go. But he does let Laertes go. When Ophelia confesses to Laertes that she likes Hamlet, Laertes says she should by no means attempt to be romantic with him. Laertes gets everything Hamlet wants and actively attempts to take away what little Hamlet has. As a symbol, Laertes is an ironic anti-hero. He does the opposite of what is good for Hamlet while getting what Hamlet wants.
The confusion about the ghost
Hamlet wants to communicate with the ghost of his father, especially since Horatio says he already has. The ghost finally appears to tell Hamlet that in fact, the father was murdered by Claudius. This means that Claudius needs to die for his sins so that Hamlet can take his rightful place as king. There's just one problem; Hamlet is left with this ultimate irony—he can't be sure the ghost was real, or if it was real, whether it is trustworthy. This knowledge, ironically, doesn't help him to be decisive.
The ironic use of art
Hamlet feels that by staging a performance of The Murder of Gonzago, he might be able to discern Claudius's guilt. He does this, but Claudius simply leaves at the moment of the murder. Hamlet takes this as evidence, but the layered use of art is complicating. Instead of gaining clarity, the argument could be made that he is sinking into paranoia. This is what the family begins to think after the play when Hamlet starts acting disturbingly.
Death as the ultimate stake
The stakes of this play are high. Hamlet must act, but what should he do? Over time, he gains clarity about how evil Claudius really is, and he hates the man for sleeping with his mother (literally), but this isn't quite enough to make him willing to take life. Then, when his true love dies, and he passes the skull of Yorick the jester, he comes into conflict with Laertes—they are both poisoned, and only then, when he is dying, is he willing to execute judgment. Ironically, it doesn't really matter, because no one gets the throne—they all die, and Prince Fortinbras is named as heir. Hamlet's role as prince is meaningless and still epic.
