Old English clæne "free from dirt or filth, unmixed with foreign or extraneous matter; morally pure, chaste, innocent; open, in the open," of beasts, "not forbidden by ceremonial law to eat," from West Germanic *klainja- "clear, pure" (source also of Old Saxon kleni "dainty, delicate," Old Frisian klene "small," Old High German kleini "delicate, fine, small," German klein "small;" English preserves the original Germanic sense), perhaps from PIE root *gel- "bright, gleaming" (source also of Greek glene "eyeball," Old Irish gel "bright"). But Boutkan doubts the IE etymology and that the "clean" word and the "small" word are the same.
"Largely replaced by clear, pure in the higher senses" [Weekley], but as a verb (mid-15c.) it has largely usurped what once belonged to cleanse. Meaning "whole, entire" is from c. 1300 (clean sweep in the figurative sense is from 1821). Sense of "not lewd" (as in good, clean fun) is from 1867; that of "not carrying anything forbidden" is from 1938; that of "free of drug addiction" is from 1950s. To come clean "confess" is from 1919, American English.