Hamlet, in director mode, tells the actors how he wants them to perform the play. He'd like it to come off naturally, which means they shouldn't be too loud, or gesticulate (make gestures) too much, as bad actors often do. Instead, they should use their discretion to build up suspense with their actions.
Most importantly, they shouldn't be tempted to get a cheap laugh, since the audience might drown out the important parts.
(Note that Hamlet gives directions as though he has some familiarity with acting himself. Hmmmmmm.)
Time for the show!
As everyone gets settled, Hamlet pulls Horatio aside, and says he's among the best men that Hamlet has had the fortune of knowing, and BTW he needs a favor: he needs him to watch Claudius's reactions to the play, especially during the scene that reenacts the killing of the King in exactly the way Claudius would've killed King Hamlet.
Together, they can figure out whether Claudius really did kill King Hamlet.
Sure, says Horatio, and then it's time for Hamlet to run off and act like a crazy duck again.
As Claudius settles in, he asks Hamlet how he's doing. Hamlet says he eats as well as a chameleon (creatures that were thought to live on a diet of air). Further, he plays on the notion that the air is full of promises of the time when he (Hamlet) will be on the throne (foreshadowing Claudius's death).
After brutalizing Claudius, Hamlet moves on to Polonius and Ophelia.
This should be a fun afternoon.
He starts flirting with—well, really harassing—Ophelia, asking if he can lie in her lap, and then pretending to be shocked when she asks, "Um, did you just suggest we do the deed in front of my dad?"
This scene is full of dirty jokes and Elizabethan euphemisms, like "nothing" as slang for "vagina."
Anyway, Ophelia tactfully demurs, telling Hamlet it's nice to see him so happy.
Hamlet quips that there's no way he could be unhappy. After all, his dad's only been dead two hours, and his mom seems quite happy. Ophelia points out that, actually, his "two hours" is more like four months. He responds that it's amazing how a great man can die and not be forgotten in two months' time.
Hamlet then plays around, talking about a hobby-horse not being forgotten. This is Shakespeare's own playful reference to the fact that regular Elizabethan village people were pretty bummed out by the puritanical suppression of sports at pagan festivals, which often had fun stuff like costumed horses and dancing.
Okay, now it's really time for the show.
The play the actors perform is a variant of "The Murder of Gonzago." In the first scene, the Queen repeatedly swears to her husband (the King) that she will never remarry. Oh, sure, says the King in the play: she's faithful now, but she'll forget all her faithfulness as soon as she's in her new husband's bed, which should happen roughly about the time her old husband dies.
This is obviously offensive to Gertrude, but she still keeps her cool.
"The lady doth protest too much, methinks," she tells Hamlet. Hamlet says the play, called "The Mouse-Trap," is a wicked piece of work, but wouldn't bother anybody with a clean conscience.
Here come the fireworks. The husband/King is taking a nap when his nephew sneaks in and pours poison in his ear —exactly what Claudius did to Hamlet's father.
Seeing this, King Claudius gets out of his seat and rushes out of the room. Sold! Hamlet has proved Claudius's guilt —to himself.
As everyone but Hamlet and Horatio rushes out of the room, Hamlet gloats about this brilliant performance.
But in come Rosencrantz and Guildenstern, with the message that Gertrude is upset about the play and wants to talk to her son.
Hamlet just accuses them of manipulating him, and so Polonius tries to convince him—unsuccessfully.
In the end, Hamlet dismisses everyone to have a little soliloquy about what's going on in the dark corners of his mind. It's nighttime, the time when "churchyards yawn and hell itself [breathes] out / Contagion to this world" (2.2.389-90).
Hamlet's feeling so good and cruel he could drink blood, but he's a little worried: he hopes that his firm bosom won't ever give way to the soul of Nero, a Roman emperor who killed his own mother. (Remember, the ghost told him not to carry out any physical punishment against his mother.)