Act IV picks up rigth after Polonius death, when Gertrude frantically tells the king of what Hamlet had just done. The king commands Rosencrantz and Guildenstern  to find Hamlet and  the advisors body, after they leave he tells his wife that Hamlets misdeeds have gone on for to long and they come to the agreement of sending him to England. Moments later Rosencrantz and Guildenstern find Hamlet rigth after he had just hid Polonius's body, after much preasure Hamlet reveals where the corpse is being hidden but in the exchange loses all type of trust on his old friends that are now taking him to meet the king and his punishment. 

Now on a boat to England, Hamlet will manage to obtain the royal letter form the king, to the king of England, where he is gonna rewrite the letter to be in his favor, by that I refer that  Hamlet will kill off his escorts (Rosencrantz and Guildenstern) instead of what the king had intended to do to him. He will also write about how a ship of pirates returned him home to Denmark. 

After that big ordeal is happened Claudius and Gertrude are discusing their  sympathies for Ophelia, who had just lost a father. Soon after Ophelia appers delusional and crazy, she had gone insane because of what had just happened to her father and also the she fact that her lover did it. In this depiction of lunacy she mentions how her brother came back from France and is furiously gathering the people to revolt against the king because of the past actions, also coincidently they start to hear the people being lead by Laertes (Ophelia's brother) aproching to the door, when they manage to break in the king knowing that he would lose that figth tries to solve the problem diplomatically by telling Laertes that Hamlet is on his way back from England, and he will be honored to take his revenge on Hamlet himself after what he had done to his father. Laertes calms down and resolves the riot he had started, on the condition that he woukd  kill Hamlet.

Now act IV ends right after the whole situation with Laertes is resolved, but before you think nothing else can happen everyone is informed of the tragic news that Ophelia, because of her lunatic state, had drowned herself in a river, which is a focal point in the story, because it will lead to the grand finale.