1560s, in drama, "person who speaks the prologue and explains or comments on events on stage," from Latin chorus "a dance in a circle, the persons singing and dancing, the chorus of a tragedy," from Greek khoros "round dance; dancing-place; band of dancers; company of persons in a play, under a leader, who take part in dialogue with the actors and sing their sentiments at intervals."
The Greek word is of uncertain origin, because the original sense is unknown. Perhaps it is from PIE root *gher- (1) "to grasp, enclose," if the original sense of the Greek word is "enclosed dancing floor," or *gher- (2) "to like, want," if the original notion is "to rejoice."
Extension from dance to voice is because Attic drama arose from tales inserted in the intervals of the dance. In Attic tragedy, the khoros (of 12 or 15 (tragic) or 24 (comedic) persons) gave expression, between the acts, to the moral and religious sentiments evoked by the actions of the play. English 16c. theater adopted a stripped-down version of this.
When a Poet wished to bring out a piece, he asked a Chorus from the Archon, and the expenses, being great, were defrayed by some rich citizen (the khoregos): it was furnished by the Tribe and trained originally by the Poet himself [Liddell & Scott]
The meaning "an organized company of singers" is from 1650s. Meaning "the refrain of a song" (which the audience joins in singing) is from 1590s; that of "a song to be sung by a (large) chorus" is from 1744. Meaning "main part of a modern popular song" (as distinguished from the verse, q.v.) is by 1926, originally in jazz. As a verb, 1703, from the noun. Chorus girl "young woman who sings and dances in a stage chorus" is by 1852.