They serve a more serious purpose. They are used by the King to spy on Hamlet, that's hardly comic relief. (Even though their fate was funny and well-deserved)
Hamlet fakes death warrants for Ros and Guil which are acted on by the King of England. Hamlet justifies this by saying that they were spies and got what they deserved, especially as this was the death which Claudius had planned for Hamlet with Ros and Guil's help. "They did make love to this employment" says Hamlet. Horatio is not wholly convinced, especially since R and G probably had no idea what their warrant said (it was sealed, remember). "So Rosencrantz and Guildenstern go to it" says Horatio, a bit disgusted by Hamlet's callousness.
I think in a duel with Ophelia's brother. But Ophelia's brother died too, because of a poisoned sword. Maybe i should start from the beginning. Ophelia's brother chalenged Hamlet. Hamlet's uncle saw this as an oppurtunity to get rid of Hamlet. So he got Ophelia's brother to put poison on his sword. And just un case that didn't kill him, Hamlet's father also put poison in Hamlet's drink. But then Hamlet's mother, Gertrude, tried to drink the poison in a toast to her son. Hamlet's uncle tries to stop her, but it's too late. Meanwhile, the swords are switched in battle. Ophelia's brother dies, but forgives Hamlet in his last moments. When Hamlet's mother dies, Hamlet's uncle is revealed to be the murderer. Hamlet challenges him to a duel. They both die.
Act 3 is a busy act in Hamlet. Rosencrantz and Guildenstern make a report to the king, and he and Polonius plan to spy on Hamlet. They place themselves in a hidden space with Ophelia as bait. Hamlet strolls along and makes his "To be or not to be" speech. He sees Ophelia and they have a very peculiar conversation which ends with Hamlet getting angry with her and accusing her of prostituting herself (figuratively). Ophelia thinks he has lost his mind ("O what a noble mind is here o'erthrown") but Claudius draws the opposite conclusion. That's Scene 1. In Scene 2, Hamlet meets up with the players, and after telling them how to do their job, asks Horatio to watch Claudius during the play. The play is then performed with Hamlet being terrifically rude both to the actors and to his fellow audience members. Claudius spots that the play is all to similar to his murder of Hamlet Sr. and demands that it stop. As he exits, Hamlet is exultant: the Ghost has been vindicated! Rosencrantz and Guildenstern show up and tell Hamlet to visit his mother. On the way, Hamlet spots his uncle praying. We hear part of the prayer and know that Claudius is feeling guilt for murdering his brother. Hamlet is about to kill him then hesitates, deciding to wait until Claudius is doing something sinful. Nevertheless he says he could "drink hot blood" and he is now going to give his mother a talking-to. In Scene four, Hamlet arrives at his mother's bedroom and begins chastising her. Polonius, hiding behind a curtain, hears and echoes Gertrude's cries for help. Hamlet, believing his moment to kill Claudius has come at last, kills Polonius thinking that he is the king. He then launches into a long tirade to his weeping mother, and is only stopped by the appearance of the Ghost. Gertrude is persuaded that Claudius is a murderer and agrees to help Hamlet, and Hamlet drags the corpse of Polonius off.
Polonius is killed when Hamlet is arguing with his mother Gertrude and Polonius is hiding behind an arras. When his mother calls for help, he also calls and Hamlet stabs him through the arras beliving that it is Claudius hiding behind it. Rosencrantz and Guildenstern are recruited by Claudius after Hamlet kills Polonius to escort Hamlet to England along with a letter for the King of England to have him killed. Hamlet finds and rewrites the letter instructing them to be killed instead. Their ship is attacked and Hamlet returns to England leaving them to go to their deaths in England. Ohphelia, Hamlet's girlfriend, commits suicide when she discovers that Hamlet killed her father Polonius. Gertrude drinks from a poisoned cup that Claudius had prepared for Hamlet to kill him. Laertes takes part in a duel with Hamlet and scratches him with a poisoned sword - poisoned by Claudius. However, later in the duel the swords are switched and Hamlet stabs Laertes with the poisoned sword, killing him. Claudius is forced to drink poison by Hamlet for revenge of killing both his mother and father. Hamlet dies after being scratched by the poisoned sword.
Hamlet didn't write any letters to Laertes--ah, you mean "Did Claudius read the letter Hamlet wrote to Claudius out loud to Laertes?" I confess that it is difficult to phrase this question so it is unambiguous. Yes, Claudius reads the letter he got from Hamlet out loud to Laertes and asks whether Laertes can explain it.
In Act 3 Scene 1 of Hamlet ("The Nunnery Scene"), Ophelia has been sent to help Claudius and Polonius spy on him. At some point Hamlet realizes this and tests her. Many productions show this as happening with Hamlet's line "Where is your father?" Hamlet knows he's watching and wants to see what Ophelia's reaction will be. Ophelia can't give the honest answer ("He's behind the curtains over there") and so replies "At home, my lord" which is a lie. Hamlet knows it, and Ophelia knows that Hamlet knows it. Hamlet knows that she cannot be trusted, and becomes violently angry at her betrayal of their relationship. Some people would put this moment earlier in the scene, at Hamlet's line "Are you honest?". Indeed, he may have made up his mind about her when he burst into her closet with his stockings down-gyved to his ankle. Rather than "the very ecstasy of love", his behaviour may have been wondering whether he can trust Ophelia, and concluding that he cannot. The Nunnery scene is the second of three scenes where the king uses those nearest and dearest to Hamlet to spy on him. In the first, his school chums Rosencrantz and Guildenstern are sent to sound him out. Hamlet suspects that they are spies, possibly because of something he sees or hears just before the line "But in the beaten way of friendship, what make you at Elsinore?" (I recommend watching this scene in the 1964 production of Hamlet with Richard Burton, to see this clearly) In the third, it is his mother who is used, but Polonius the spymaster gives himself aways and pays with his life. Spying is a constant theme in Hamlet, which is nicely pointed up in Grigori Kosintsev's 1964 film and David Tennant's 2010 production.
The character of Hamlet's girlfriend who goes crazy is called Ophelia. It's Greek for help.
Hamlet is killed in Act 5, scene 2 when Laertes cuts him with a poisoned blade. Hamlet's Uncle Claudius has convinced Hamlet to duel Laertes and has supplied Laertes with the poisoned blade that kills Hamlet.
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Horatio
Hamlet stabbed him right behind the arras in Queen Gertrude's room.