to do away with entirely: to kill a person, or abolish an institution
The root idea behind this word is taking something up and away, so its dominant use is doing away with a person, that is, putting someone to death. In Acts especially it is the standard word for lethal violence: Herod's slaughter of the infants, the repeated plots to kill Paul, the execution of James, and the stoning of Stephen all use it. It does not by itself name a method, whether sword, stoning, or formal execution; it simply says that someone was killed.
The same idea of removal reaches beyond people to things. In Hebrews the word describes God taking away the first covenant order so that a second can stand in its place. The object there is an institution, not a person, but the underlying action, doing away with something completely, is the same.
Because Luke supplies most of the New Testament uses, the word carries the flavor of his narratives of persecution and murderous intrigue, where it is almost a routine term for a life being taken.