To UNDER-expose the background, the easiest thing to use is a softbox that has a so-called "egg-crate" grid on the front of it. A softbox tends to fall-off more rapidly than an umbrella, and ther inexpensive, made-in-CHina softboxes that have the "egg crate grid" accessory make the light fall-off in intensity VERY rapidly, and keep the light OFF of the background of whatever photography set they are use on.
So--if you can, use a softbox, and if you have one with an eggcrate grid, definitely use that to keep the light so it stays "up front" on the set, and the backdrop drops off in brightness very rapidly.
SECOND point: move the subject at least SIX feet away from the bak wall. THEN, keep the main light CLOSE to the subject. With a SHORT distance from the light to the subject, the light is soft--but it also falls off in intensity very,very quickly. If you move the light farther away from the model and then open the lens wider from say f/11 to f/5.6 to compensate, the background will be brighter than it was in earlier shots--even though the camera-to-subject distance stayed the same, and the subject-to-background distance stayed the same!
SO: subject six to ten feet from background: main light CLOSE to subject, to make backdrop appear dark. A PLAIN, white wall can be made gray to dark gray this way, under normal room conditions.
Target stores sell nice microfiber blankets in queen and king size, in a nice, medium gray color. I like these as portable backdrops, but you need "some way" to hang them up. A crossbar of pipe, and some A-clampos will work. Seamless paper on 9-foot wide rolls uses a similar system to hang it up..but YOU CAN ALSO unfurl the end of the roll t"upward", leaving the main roll on the ground , and then TACK UP or tape up the end of the roll to the wall!