The Third Rome claim comes from the fact that the niece of the last Emperor of Byzantium married the Grand Prince Ivan the Great of the ancient kingdom of Muscovy and thereby gained the right to use the regalia of the Caesars including the Imperial Standard, the double headed eagle carrying the Christian cross surmounted orb of rightful rulership and a sword, and the right to the the title of Tsar or Czar (Caesar or Emperor)

The Byzantine world view was that kings ruled over countries (there were twelve different thrones involved in the Russian Empire) and emperors ruled over kings. Hence the title King of Kings.

Remember, according to Revelations, He is the Prince of the Kings of the Earth.