What did Fortinbras do in Act 1 Scene 1 that alarmed the powers that be in Denmark?
Fortinbras' threat to Denmark is the main focus of the opening of the play. His family position is similar to that of young Hamlet. Fortinbras' father was King of Norway and his uncle now holds the throne. Hamlet's father was slain and now his uncle Claudius (also stepfather) has the throne of Denmark. King Hamlet had defeated Fortinbras' father in battle, and Fortinbras is now seeking to avenge the defeat and have the lands lost by Norway returned to him. This position is explained by Horatio-
Now, sir, young Fortinbras,
Of unimproved metal hot and full,
Hath in the skirts of Norway here and there,
Shark'd up a list of lawless resolutes,
For food and diet to some enterprise
That hath a stomach in't; which is no other—
As it doth well appear unto our state—
But to recover of us, by strong hand
And terms compulsatory, those foresaid lands
So by his father lost.
Fortinbras' army are made to sound like fearful mercenaries.
Fortinbras is angry at Denmark because Hamlet's father had killed his own father. Fortinbras's father had been the king of Norway. So now young Fortinbras wants revenge. What he is doing that is alarming the Danish government is threatening to invade the country.
Specifically, young Fortinbras has been hiring himself an army in Norway (Horatio says it's a bunch of criminals that Fortinbras has hired and has paid in food). It looks like he's going to have them invade to take revenge and also to take back some land that Denmark won from Norway.