Hamlet Act I, Scene v Summary

And now for the big money talk: The ghost (who says he is indeed the ghost of Hamlet's father) says he needs revenge, and sooner rather than later, since he's doomed to burn in eternal flames until he gets it. (The ghost appears to be suffering in Purgatory, a space where sins were "purged" before a spirit could reach heaven. Check out the summary for Act I, Scene iv for more on how people viewed ghosts during the Elizabethan period.)
The ghost reveals that Claudius, his brother, the one now married to Gertrude, is the guy that murdered him.
But how? The ghost reveals the details: While he (King at the time) was sleeping in his orchard, Claudius snuck over and poured poison in his ear.
Hamlet promises to avenge his father's death. Let's all take a moment to write this down. Hamlet promises to avenge his father. He does, no fingers crossed behind his back or anything. This promise has huge implications for the rest of the play.
The ghost is pleased (as much as a ghost can be), but he warns Hamlet not to do anything to Gertrude, his mother.
Hamlet, returning to the others, swears his friends to secrecy. He also warns them that he might start behaving strangely, but they shouldn't freak out because he's just acting on instructions given to him by the ghost of his dead father who told him of an evil and murdering plot for the crown. Specifically, he says he's going to put on an "antic disposition," which means he's going to pretend to be mad, like an "antic" (which is the role of a theater clown or a grotesque).
Also, everyone swears to not tell anyone anything.
The ghost's disembodied voice adds that they had really better be swearing.

Act II, Scene i

Act I, Scene iv