Why didn't Prince Hamlet become king after the death of his father? How is it that Claudius became king?

Primogeniture was not practiced in Shakespeare's day. Although King Hamlet had a son, that son's succession to the throne was not automatic. Perhaps the King had not yet named his heir, which would open the kingship to election by the nobles. There is evidence of this in Hamlet's comment following the death of Claudius and his own fatal wounding:

I do prophesy the election lights on Fortinbras.

It is likely that after killing King Hamlet, Claudius used Hamlet's absence at school and the fact of his marriage to Gertrude to manipulate that election so that he could be king. He was certainly a skilled diplomat, and there was definitely an immediate need for rulership, given the impending threat from Norway. It seems very likely that Claudius used this pretext to manoeuvre his way onto the throne.