do away with

Very likely a continuation of Old English onweġ dōn (“to get rid of”, literally “to do away”, from onweġ (“away”) and dōn (“to do”, also “to make, cause, put”)), via a Middle English *awey don, *oway don, *don awey, and other forms. The preposition with was probably also already in use from the outset, given how its original meaning is “against” (still seen in e.g. argue with, withstand and have a fight with). The original meaning of the phrase was therefore on the lines of “to put away against (someone or something)” or “to make/cause (s.o./sth. to go) away”, which are long-obsolete senses of do.