fill (v.) Look up fill at Dictionary.com
Old English fyllan "to fill, make full, fill up, replenish, satisfy; complete, fulfill," from Proto-Germanic *fulljan "to fill" (cognates: Old Saxon fulljan, Old Norse fylla, Old Frisian fella, Dutch vullen, German füllen, Gothic fulljan "to fill, make full"), a derivative of adjective *fullaz "full" (see full (adj.)). Related: Filled.

To fill the bill (1882) originally was U.S. theatrical slang, in reference to a star of such magnitude his or her name would be the only one on a show's poster. To fill out "write in required matter" is recorded from 1880.
fill (n.) Look up fill at Dictionary.com
mid-13c., fille, "a full supply," from Old English fyllu "fullness, 'fill,' feast, satiety," from Proto-Germanic *full-ino- "fullness" (cognates: Old High German fulli, German Fülle, Old Norse fyllr), noun of state from *fullaz "full" (see full (adj.)). Meaning "extra material in music" is from 1934.