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Hamlet doesn't. Claudius does. Their names are Cornelius and Voltimand.
One of the two ambassadors Claudius sends to Norway. (The other is Cornelius.)
Fortinbras
Claudius deals with Laertes first, before turning to Hamlet. He asks why he is still depressed. Hamlet, after a discussion with the Queen, makes a speech about how genuine his grief is. Claudius then tells him that he is mourning too much and should stop, that he is the heir to the throne, and that he won't be leaving Denmark to go back to university.
guard
King Hamlet won, and therefore won certain territories in Norway, which young Fortinbras resents.
Basically, Norway wins. Fortinbras is a citizen of Norway, and he takes over Elsinore castle at the end.
Fortinbras of Norway takes the throne after Claudius and Hamlet die
Fortinbras had it coming because his name was gay
Nope, at least the Norway government hasn't to stop funding terrorist organizations. On the other hand, he's a pretty well-rounded fellow and has probably read or seen Hamlet at some point.
I suspect Claudius does not take young Fortinbras as a serious threat to Denmark. At Act 2 Scene 2, news of Norway from Voltemand would appear to be "good news", yet this diplomatic importance to the King plays second fiddle to Hamlet's health. Claudius does not promptly respond to this message and would wait/hesitate (much like hamlet) without taking prompt action: "Wasting time", as a nervous Polonius would interject The message when looked at closer from Voltemand's speech, suggests that Old Norway had arrested his nephew, briefly, yet given him 3000 crowns to employ his army previously levied. This is basically taking money out of the left pocket and putting it in the right pocket. I suppose a better question would be, "Should Claudius fear young Fortinbras?" Claudius publicly states to the court that young Fortinbras holds an arrogant view of Denmark, crippled by a pyrrhic victory. Instead of answering this young man directly, Claudius turns his attention to Old Norway and sends "dilated articles" (detailed instructions) on how to handle young Norway. I do wonder if those 3000 crowns was of Claudius' money. The fortified garrison watch would suggest so, yet no offensive manoeuvrings are planned from Denmark. Diplomacy seems to be the best course of action for the treacherous King. Does Fortinbras intend to conquer Denmark? Or reclaim the land lost by his father in battle in Poland? Does he come to Elsinore directly from Norway's lands, or on his way back from Poland? Where are the soldiers levied?
He ends up being King of Denmark. Since the entire royal family of Denmark gets dead in the last act, and Fortinbras happens to be there with an army, he wins by default. Fortinbras says that he has some claim to the throne of Denmark. This is crap--he has no claim at all as Claudius showed clearly in the first act. He is just a scumbag adventurer who happens to be in the right place at the right time.