In the play, Hamlet, who named Hamlet heir to the throne?

This is actually a tricky question because no one actually
named young Hamlet as heir to the throne.  He is never actually the King.  In the time
of the play, Denmark is an elected monarchy.  This means that it is not an automatic
event that the first born son of the monarch will be king.  This is how Claudius gets
the throne after King Hamlet's death.  While we don't know what exactly happened in the
immediate aftermath of King Hamlet's death, it is clear that Claudius made a case to the
royal court that he should take the throne.  He may have suggested that Prince Hamlet
was too young, too inexperienced, still attending school and therefore too removed from
the day-to-day leadership of the country.  No matter what, Claudius does acknowledge the
court's role in putting him on the throne when he thanks them "for going with this
affair along."

In Act 5, after Claudius dies, Hamlet is the
next logical royal subject to ascend the throne.  In all likelihood he would be elected
king if he had lived.  Hamlet knows that he has been mortally wounded and in a "kingly"
act says, "I do prophesy the election lights / On Fortinbras.  He has my dying voice." 
He is ensuring that the throne of Denmark pass to a suitable leader -- a member of
royalty -- who also impressed Hamlet with his drive and determination earlier in the
story.  In these lines you see the reference to the "election lights" reminding the
audience that the royal court will have to affirm the next king through an election, but
we are to assume that Hamlet's request would naturally be
honored.