Polonius tells his servant Reynaldo to spy on Laertes by speaking to Danes in Paris about his whereabouts, and the reputation he upholds. Polonius gives Reynaldo consent to lie in order to bring out the truth.
ACT II, SCENE 1, LINES 3-23
I want it done even if you have to lie.
Ophelia comes rushing in while Reynaldo exits. She tells Polonius about her indecent encounter with Hamlet. Polonius has no other reason but to believe that Hamlet's love for Ophelia has made him "mad". After being told, he goes to the king with this information.
he was "without a hat on his head, his shirt unbuttoned, and his stockings dirty, undone, and down around his ankles. He was pale as his undershirt, and his knees were knocking together."
Hamlet needs to stay away from Ophelia. He is MAD!
ACT II, SCENE 1, LINES 77-107
King Claudius and Gertrude call on Hamlet's old friends, Guildenstern and Rosencrantz, to help keep an eye on Hamlet. They agree to the king and queen's terms, and are promised royal compensation in the near future.
ACT II, SCENE 2, LINES 1-40
Do this for us and your dear old friend Hamlet.
Hamlet is not "mad" because of Ophelia...
As Polonius exits, Gertrude has a hard time believing that Hamlet's "madness" is because of his affection for Ophelia. She believes it is driven solely from his father's death, and her early marriage. Polonius then strikes the idea of hiding behind the arras with King Claudius to truly see if Hamlet's madness is driven by Ophelia.
Let's hide and eavesdrop behind the arras... We'll see who's right then.
ACT II, SCENE 2, LINES 56-57 & 163-168
Oh no... he's onto us...
Hamlet is encountered by Guildenstern and Rosencrantz. He quickly sees through their lies and asks them their true intentions as to why they're in Denmark. Guilty, they say the king and queen has put them up to the task of spying on him. Luckily, Rosencrantz breaks through the awkward moment between the them by announcing the players' arrival. As they enter, Hamlet requests they perform Virgil's Aeneid.
Tell me the truth now!
ACT II, SCENE 2, LINES 261-284 & 355 & 431-445
As the players exit, Hamlet reveals that they will perform a play called The Murder of Gonzago tomorrow night, which will (hopefully) speak to King Claudius' conscience, guiltily proving that he murdered the king. Hamlet will observe his emotional reaction, and see whether he confesses to the murder or not.
This play will reveal the truth of my father's death to me!