Is the word king a common noun?

Yes, the noun 'king' is a common noun, a general Word for any king.

A proper noun is the name of a specific person, place, or thing. The noun 'king' is a proper noun as the name of a specific person such as King Henry VII or Carole King (Americn singer/songwriter); the name of a place such as the town of King of Prussia in Pennsylvani; or a company such as Burger King.