There are a couple of key myths referenced in Hamlet. The most notable are the figures of Hyperion, Pyrrhus, and Hecuba.
In act 1, scene 2, Hamlet references the sun god Hyperion and compares him to his father. He says "so excellent a king, that was to this Hyperion to a satyr" (1.2.143-144). This is a short reference to a myth, but the comparison is used to express how perfect Hamlet believes his father was. He compares him to a sun god, but compares Claudius to the mischievous satyr, which is a creature that is half man and half goat.
In act 2, scene 2, when Hamlet encounters the players, he asks them to re-enact the story of Pyrrhus. This is significant, because in Virgil's Aeneid, Pyrrhus is a vengeful son of the warrior Achilles. Pyrrhus takes revenge on Priam and hacks him to death as revenge for the death of his father. This is significant in Hamlet because Hamlet is also seeking revenge for the death of his father.
After the players deliver the speech about Pyrrhus, they also describe the grief of Hecuba. This is significant for two reasons. Hecuba was Priam's wife, and she grieved wildly for him after his death. This is noticeably different from how Gertrude is behaving. Secondly, the actor playing Hecuba appears to genuinely be moved by extreme grief, and Hamlet is once again amazed at how an actor can feign such passion and grief, but he himself cannot bring himself to avenge a real situation.