"Because you are the true King of Narnia; Caspian the Tenth. The true son and heir, long life to your Majesty."
In fact, Caspian was a Telmarine. The Telmarine's conquered Narnia and probably killed the actual King they had at the time. In the Magicians Nephew, Aslan says that the sons and daughters of Frank and Helen would be kings and queens in Archenland.
"Rise up King and Queen of Narnia, father and mother of many kings that shall be in Narnia and the Isles and Archenland."
Because of this, I think the true king of Narnia would be one of the relatives of the king in Archenland at this time. There is no indication in the books that Archenland was conquered as well by the Telmarines.
Why would Dr. Cornelius say that Caspian is the true king of Narnia if he is an old Narnian? I would think that he would want a descendant of Helen and Frank to be the king of Narnia. I wonder that no one else in the books tried to get to Archenland to get a descendant to reign over Narnia.