As Wayne says you don't need to overexpose the background, however it does need to be illuminated properly. A single flash illuminating both may not do the job correctly, plus you don't want the subject too close to the background as you may get some "Light wash" spilling on to the subject.
I would light the subject normally then use a background light to give about 1 stop more exposure to the background . This ensures the background is well lit but not blown out.