In ancient Greek drama, the Chorus served a number of purposes. It would comment on events taking place in the action, offer insight, provide background information on characters and situations, provide clues to characters' real thoughts and feelings, or even offer clues as to play's themes and meaning. By including a Chorus in this play, Shakespeare seems to raise the significance of Romeo and Juliet's story to that of myth, to that of classical theater. By employing a Chorus, he seems to rank their tragedy with the likes of Oedipus Rex, Dido and Aeneas, or Medea, for example.
In addition, Greek audiences would have already known the story of, say, Oedipus, before they came to see the play. The skill of the author was not determined by his ability to tell a good and original story, but to tell a well-known story well. The creation of dramatic irony was paramount to his endeavors, building tension for the audience as they wait for the characters to finally learn the truth that the audience, themselves, already knows. Then, once the truth is out, the audience experiences the relief of catharsis, a sort of emotional purging of the tension that's been sustained throughout. By having the Chorus spoil the end of the play before it even starts—Romeo and Juliet die, and only then do their families end the feud—Shakespeare sets up similar possibilities for the development of irony, tension, and catharsis here. This could also be interpreted as an attempt to elevate the story of this couple to be on a level with classical tragedy.