What MrBob (and Copperfox) said. It looks like Miraz was acting as regent for Caspian--which means, essentially, that he ruled Narnia while Caspian was too young to do anything. In the book that makes sense, because Caspian is still like, twelve, but in the movie Miraz should have stepped down much earlier, or Caspian/Caspian's supporters should have firmly seized the throne. But as regent, there was no reason for Miraz to take the throne--he was in power until Caspian was "old enough" to rule. It was only the birth of Miraz's son that caused a problem--there was no way Miraz could have handed the crown to his son.
Of course, Miraz trying to kill Caspian THAT VERY NIGHT was a little silly. The smart thing would have been to wait a little and then have Caspian die in a hunting accident or something.
I do wonder why Caspian never tried to seize the throne and why he never seemed to think to himself "Why is it that my father was king, and I am thirty, and yet Uncle Miraz never lets me do anything?" OTOH, it's not like movie!Caspian was much good at politics or anything at any point in the movie, so maybe that's in character.