With the dark BG you do need some light to separate the hair from the BG, a weak spot from behind or one side usually works. Sometimes a gobo or barndoors is needed to keep the light from from spilling over. Also you might try one dead behind the head to create a bit of a halo from her hair. Have fun and show us what you get!
My preference if you don't have to have a specific background is to use a Burgundy Red as the background. I find in B&W it comes out a nice mid-grey that can be used with any colour hair. If your after a dark background I'd go with Gary's recommendations to separate them.
This background's ok, but you need a hair light. I use a snoot cone for this, to keep it from adding flare. Illuminating the background just a little wouldn't hurt, either, but it's not necessary if you put a spot on the top of her head.
Forget the background, that thing on her lip makes me wince every time I see this photo. I tried waiting before I posted and come back and see it a couple of times...same thing, that stud on her lip is just too distracting for me. Of course I realize I am too conservative and that the "been tagged by the wild life society" look is supposed to be in fashion....boogles my mind....anyhow. I agree perhaps a reflector on the top of the head would have given you a little more separation. Other than that I like the pose and tonality.