What is the significance of Shakespeare using the theme of appearance vs. reality in Hamlet? Why does he use it? How does that tie into the overall...

The overarching problem—the key problem—facing Hamlet is determining the difference between appearance and reality, and it is an issue Hamlet continuously interrogates. From his first encounter with the ghost, Hamlet faces one fundamental problem: is the ghost really the spirit of his dead father, or is it a demon sent by Satan to deceive him into murdering an innocent man? How can he determine who around him is telling the truth and who is lying? More specifically, how can Hamlet determine if Claudius is responsible for the death of Hamlet's father? What, Hamlet keeps asking, is real? What is appearance (a lie)?

Deception permeates the play: Polonius and Claudius hide behind tapestries to spy on Hamlet. Claudius sends Rosencrantz and Guildenstern to England with secret orders to have Hamlet killed. Hamlet watches in wonder as one of the actors in the travelling drama troupe fakes emotion. When Ophelia dies, Hamlet mocks her mourners and accuses them of false emotion.

Hamlet's Denmark is "rotten," a corrupt place, and Hamlet recognizes that lies (appearances) are at the center of the rot. Uncovering what is real is essential to Hamlet's mission of avenging his father's death, because until he establishes what is true, he cannot ensure justice is being served.