The King Fish is an unnamed character that only appears in the episode "Sleepy Time." He only appears during Squidward's dream sequence.
King Fish is a green fish that wears a crown and a cape, the traditional clothing articles of a king.
When dream SpongeBob invades Squidward's dream, Squidward is playing solo with his clarinet in front of a crowd. When he is interrupted by SpongeBob, the King Fish makes his first appearance and tells Squidward to keep playing the music. Squidward starts playing again but is once again interrupted by SpongeBob's laughter but notices him with the King Fish. King Fish once again yells at Squidward to keep playing and he continues to talk with SpongeBob. Squidward gets so angry that he accidentally breaks his clarinet. King Fish tells Squidward that if he doesn't hear any music soon, it will be his head. Squidward tries to find a solution until SpongeBob appears next to him turned into a clarinet. Squidward declines but an "Ahem" from King Fish reverses his decision. Squidward plays the SpongeBob clarinet which sounds horribly to him and stops playing. However, the King Fish loves the music and begs Squidward to keep playing. Squidward does until the crowd knocks him away picking up SpongeBob who then runs off. King Fish, along with the crowd, chases after SpongeBob as he laughs.
King of Bikini Bottom actually shouldn't exist, because Bikini Bottom is a town, not a kingdom.