Hamlet Act V, Scene i Summary

At the palace graveyard, two witty gravediggers discuss Ophelia's death. (They vaguely reference the famous 1554 suicide case of Sir James Hales, a dead-by-suicide judge whose land was forfeited to the crown, as was standard punishment for suicides. Mrs. Hales, Sir James's wife, argued that the court could only punish someone for acts committed in their lifetime. "Successful" suicide required the ending of one's life – thus when her husband succeeded at suicide, he was no longer living and couldn't be held accountable, meaning… she should be entitled to his stuff. Hence all the quibbling on whether Ophelia meant to kill herself or not.)
These two men think Ophelia's death was clearly a suicide. Because suicide is an offense against God, those who committed the act were usually not allowed to be buried in a Christian graveyard. The gravediggers, however, conclude that it's only because her family is rich and powerful that Ophelia is getting a Christian burial.
The two continue to crack jokes while digging a hole for the young woman. One is left singing (while digging) as Hamlet and Horatio approach.
Hamlet wonders whether this is a particularly callous gravedigger, that he can sing while he works.
Horatio points out that doing a task over and over makes it easier, so this guy is well-adjusted to his work, and he might as well whistle away.
Hamlet philosophizes about how the gravedigger has thrown up two skulls from the hole in the ground. Hamlet wonders who these skulls used to be when they were men.
He marvels that man's life's work and all of his possessions came to equal exactly the same at the end – exactly nothing.
The gravedigger is a chatty guy, and Hamlet makes conversation about how long the man has been lying about in other people's graves.
The gravedigger says he's been at it since the day the old King Hamlet beat Fortinbras, which, coincidentally is the day the young Hamlet was born.
The gravedigger quibbles a bit more, not knowing his audience. He says Prince Hamlet has gone mad and has been sent to England, where it doesn't matter if he's cured since everyone there is mad, too.
Finally, Hamlet gives up on his strange line of questioning and takes up a skull handed to him by the gravedigger. The gravedigger admits the skull once belonged to a man, Yorick, the former King's jester.
Hamlet reminisces about the times Yorick used to cart him around on his back and generally makes a melodramatic scene about all the jokes and laughs and dances Yorick used to do, but now can't, being dead and all.
Hamlet asks Horatio if this is the fate of all men, great and small, to be rotten and disintegrating and otherwise dead. Hamlet particularly asks after Alexander the Great, wondering if you could trace his dust until you found him stopping up a "bunghole" (which was literally the hole in a barrel or cask of drink, but, you guessed it, has been slang for "anus" since the thirteenth century.)
Hamlet waxes on about death until he's interrupted by the King approaching with the Queen and, once again, their country entourage.
He wonders whom they're all following to the grave, and he and Horatio hide to watch what happens.
Hamlet recognizes Laertes and calls him a "very noble youth."
The priest leading the ceremony makes nasty comments about how Ophelia may have committed suicide. He says her grieving family and friends have done all of the ceremony they can do without making a mockery of people who have actually died and gone to Heaven in peace. They should be glad they were even allowed to avoid burying the girl in some awful, unholy ground where people could throw rocks at the headstone.
Laertes doesn't take kindly to all this trash talking of his recently dead sister, and says that Ophelia will be an angel while the priest howls in Hell.
Everyone has a different reaction to Ophelia's death: Laertes is still itching for a fight; Hamlet is shocked to hear the girl is dead at all; and as Gertrude scatters flowers on the grave, she laments that Ophelia didn't end up being Hamlet's wife.
Laertes declares that he wishes woe times ten to fall upon the man who did the deed that made Ophelia crazy (alluding to Hamlet's murder of Polonius).
Laertes says he must hug his dead sister again and jumps into her grave, telling the diggers to bury them both, the quick (meaning "the alive") and the dead.
Hamlet comes forth out of hiding to Ophelia's graveside. He announces, "This is I, Hamlet the Dane," which is pretty significant, as that was a way of announcing the King.
Hamlet promptly leaps into Ophelia's grave and scuffles with Laertes. The two are broken up, and Hamlet declares he'll fight Laertes "on this theme," meaning, the question of who loved Ophelia more, until the end of his days.
He challenges Laertes to a fight to settle the matter in question, claiming he's as willing to be buried with the girl as Laertes could possibly be.
Ultimately, Gertrude excuses Hamlet for his little fit and says she's sure he'll get over it soon.
Hamlet declares that every dog will have its day, ominously, and he leaves.
King Claudius sends Horatio to look after him.
Finally, Claudius has an aside to Laertes, telling him to be firm about what they talked over yesterday (the plot to kill Hamlet).

Act V, Scene ii

Act IV, Scene vii