What is the role of revenge in Hamlet? How is revenge represented? What is its dramatic function in the play?

Revenge is represented as the ultimate destructive force in Hamlet. It draws out the worst traits in the characters seeking it, and has negative consequences on bystanders.

The first act of revenge occurred two months prior to the start of the play, when Claudius poisoned his older brother to be king of Denmark. We never hear the whole story from Claudius, just that he did it for "My crown, mine own ambition, and my queen" (III.3). Claudius was clearly jealous of all that his brother had and plotted revenge to take it all. This revenge act plays perhaps the most important dramatic function, since none of the other revenge plots, with all their resulting tragedy, would develop without this catalyst.

Next, Hamlet vows revenge on Claudius once the ghost reveals the murder plot. Hamlet is not vengeful by nature, and so never directly seeks opportunity to murder his uncle. Yet all the convoluted ways he approaches the task do have negative effects on those around him, thereby furthering the plot. His cover of insanity turns all eyes on him and deeply disturbs Ophelia, which prompts her insanity. Hamlet's odd behavior also causes Polonius to poke his nose where it doesn't belong, which places him behind the arras in his Gertrude's room, which leads to his death. This, in turn, sets Laertes to seek vengeance.

Laertes' revenge looks quite different than Hamlet's, even though it is for the same cause: a murdered father. Laertes is willing to do whatever it takes to gain his revenge on his father's murder, even "cut his throat i' th' church" (IV.7), meaning that unlike Hamlet, Laertes is more than willing to damn his own soul to accomplish his revenge. Ultimately, he gives up his soul and his life. By joining causes with Claudius and rashly rushing to revenge, Laertes ends up being responsible for Gertrude's death in addition to Hamlet's, as he supplies the poison that she inadvertently drinks. 

In the end, no matter the motive for revenge, Shakespeare shows us revenge is never sweet, and definitely never without consequences to the avengers and those around the person seeking revenge. To prove this point further, Horatio, the only main character not caught up in the vengeance, is the only one to survive the play.