Why didn't the throne go to Hamlet when his father died? Isn't he the natural heir to the throne? Does Claudius have a right to assume the throne?

Shakespeare never really explains why Claudius takes the throne. Hamlet mentions "the election," which would lead us to belive that the new king has been chosen by the nobles rather having inherited the throne on the basis of lineage. It is also possible that Claudius was able to assume the kingship simply because he married the Queen. Denmark may have had a matrilineal order of ascension.