1580s, of uncertain origin, probably alteration of Middle English grim "dirt, filth" (early 14c.), from Middle Low German greme "dirt" or another Low German source, from Proto-Germanic *grim- "to smear" (source also of Flemish grijm, Middle Dutch grime "soot, mask"), from PIE root *ghrei- "to rub." The verb was Middle English grymen (mid-15c.) but largely was replaced early 16c. by begrime.
ghrēi-, Proto-Indo-European root meaning "to rub."
It forms all or part of: chrism; Christ; christen; Christian; Christmas; cream; grime; grisly; Kriss Kringle.
It is the hypothetical source of/evidence for its existence is provided by: Greek khriein "to anoint, besmear;" Lithuanian grieju, grieti "to skim the cream off;" Old English grima "mask, helmet, ghost," Middle Low German greme "dirt."
1736, in Kentish dialect, "dirty, foul," a word of uncertain origin, but perhaps related to dung. Meaning "soiled, tarnished, having a dull, brownish color" (from grime or weathering) is by 1751; hence "shabby, shady, drab" (by 1855). The noun dinge "dinginess" (1816) is a back-formation; as a derogatory word for "black person, Negro," by 1848. Related: Dingily; dinginess.
Middle English crome, crumme, from Old English cruma "fragment of bread or other food, a morsel, small fragment," from a West Germanic root of obscure origin (compare Middle Dutch crume, Dutch kruim, German Krume); perhaps from a PIE word for "small particle of bread" and cognate with Greek grumea "bag or chest for old clothes" (Beekes writes: "In origin, the word probably denoted small things of little value, later also the chest, etc.), Albanian grime.
The unetymological -b- appeared mid-15c., in part by analogy with words like dumb. Slang meaning "lousy person" is 1918, from crumb, U.S. slang for "body-louse" (1863), which were so called from resemblance.