Who was claudius? How did he marry gertrude his brother widow?

Claudius is the new King of Denmark, apparently having usurped the title and the throne from its rightful heir - Hamlet, son of the dead king. Claudius is a clever politician and manipulator, self-serving and cynically aware, in ways that Hamlet is not, of how the world works. Claudius ends up marrying Gertrude after the King is dead. It is unclear if they had a "relationship" before the King's murder.