Polonius orders Reynaldo to spy on his son Laertes to check how he is behaving. Reynaldo is supposed to do so by spreading rumors about Laertes to his friends and acquaintances. They will either deny or confirm them.
Ophelia comes to Polonius's chambers scared. As she explains to her father, Hamlet had come to her room pale, inappropriately dressed and saying weird stuff. Polonius and Ophelia assume Hamlet's behavior is the result of her rejection.
Polonius tells the King and Queen about Hamlet's behavior. He explains his reasoning to think Hamlet is heartbroken from Ophelia's rejection and reads them a love letter sent by Hamlet to Ophelia.
Rosencrantz and Guildenstern were sent by the King to check on Hamlet. Hamlet makes them tell him what was the real reason behind their visit. Hamlet starts talking deeply about his feelings, showing that he really is depressed.
Polonius sets up Ophelia to meet with Hamlet so Claudius and him can confirm that Hamlet's madness is because of Ophelia's rejection. Hamlet notices he is being spied on and decides he can't trust Ophelia anymore.
Hamlet feels weak. He frustrated that an actor is more courageous and passionate about revenge tha he is. Hamlet wants to confirm his uncle is really the murderer of his father. He decides to put on a play to see if his uncle's reaction shows any remorse or guilt.