King as a surname likely derives from an ancestor's nickname rather than an occupational byname. Nicknames derived from a variety of characteristics, including appearance, personality, and habits. So someone named King would likely have been either kingly (or notably un-kingly, for a sarcastic nickname) in personality or played the character of a king in a play. (Religious plays were staged by local people in various towns throughout England in the 14th-16th centuries, so John-who-always-plays-the-king might be a good way of distinguishing a particular person from John-who-has-red-hair and John-who-makes-barrels.)

Sources:

"Surnames and society in West Yorkshire: An analysis of names in the Wakefield Court Rolls, 1274–1352" (an MA thesis by Wendi Dunlap). Dunlap cites the entry for "King" in Origins of English Surnames by Percy H. Reaney, to which I do not have convenient access, but Reaney's academic work on names is well-respected.

"Mystery plays: a brief insight and link to prose" from the University of Cambridge Faculty of English. (https://web.archive.org/web/20160127040742/http://www.english.cam.ac.uk/medieval/mystery_plays.php)