Ah, what a fun question!
Well, assume the basics: there will be considerable confusion among those who were near power, but not in it. (Those who were in power aren't confused because they are, well, dead.) In addition, there will be mourning and dismay across the country. The royal family is dead, and there have been two kings in quick succession. Now there will be a third. Given the period's beliefs that what happened on earth was a direct result of heavenly designs, this will mean anxiety and a turn towards religion.
However, on the purely political level, Fortinbras will take control. He's there, he's legitimate, and he is both decisive and respectful. He will move firmly, but not too quickly, to secure the throne. He'll pay respects to past powers, then make a pleasantly dull reign.
Fortinbras will surely have enough problems back home in Norway. His conquest of Denmark goes directly against his King/Uncle's express instructions and diplomatic promises. He has basically guaranteed civil war in Norway too. He was publicly 'rebuked' by the King of Norway even for planning the invasion. It was forbidden. He has openly disobeyed the King.
So the King of Norway must 100% believe that Fortinbras has declared open war on him. Which way will the Norwegian Army side?
Much is still uncertain for Fortinbras's wild and headstrong military adventure. What will Poland do (the European super-power in 1600). The Poles had an army ready to oppose Fortinbras's suspected Polish invasion. Will they approve of this regime change in Denmark. Will they send this army on the offensive to stop this threatening Prince's advance? I would.
Another big question is... where the hell is the Danish Army? Fortinbras seemed to walk into Elsinore with barely a shot.
Will Laertes lead another peoples' revolt?
Will the King of Norway attack?
Will Poland attack?
What of the French?
And how does the Pope feel about all this?
And the Holy Roman emperor?
Will all of them passively accept a sudden power grab by a Norweigian upstart? Or will they sniff the opportunity and look at the turmoil in Denmark as an opportunity their own ambitions?