The Klingon Imperial Family are direct descendants of Kahless the Unforgettable. Those who are born into the Imperial Family are "of the blood", meant to rule by the divine will of Kahless.
The original Imperial Family were put to death at the end of the second Dynasty. A new group of Klingons were given their names and titles ten years later when the Third Dynasty was founded in order to create the illusion of an unbroken line. (DS9:"You Are Cordially Invited...")
The House of Kor, led by Kor, son of Rynar, was descended of the Imperial Family. (DS9: "Once More Unto the Breach") as was the House of Mogh.