dust (v.) Look up dust at Dictionary.com
c. 1200, "to rise as dust;" later "to sprinkle with dust" (1590s) and "to rid of dust" (1560s); from dust (n.). Related: Dusted; dusting. Sense of "to kill" is U.S. slang first recorded 1938 (compare bite the dust under dust (n.)).
dust (n.) Look up dust at Dictionary.com
Old English dust, from Proto-Germanic *dunstaz (source also of Old High German tunst "storm, breath," German Dunst "mist, vapor," Danish dyst "milldust," Dutch duist), from PIE *dheu- (1) "dust, smoke, vapor" (source also of Sanskrit dhu- "shake," Latin fumus "smoke").

Meaning "that to which living matter decays" was in Old English, hence, figuratively, "mortal life." To bite the dust "die, be slain, perish in battle" is from 1750, earlier lick the dust (late 14c.), which OED identifies as "a Hebraism," but Latin had the same image; compare Virgil's procubuit moriens et humum semel ore momordit.