Pup seems to be the resident expert in low-light stuff. He's taken some really great shots of the proverbial black cat in a coal cellar at night.
For example, there's an appropriately-named thread that he started titled "Photographing musicians in dark venues":
http://www.digitalcamerainfo.com/bbs/t1391.html
The gist of it was that you need a fast lens, i.e. one with an aperture that can be cranked as wide open as possible to let as much of what little light is available through to the film/sensor.
He'll likely have better recommendations for lenses than I ever would, but that's likely going to mean using a 50mm prime (which typically open up to f/1.8 for cheapos or 1/1.4 or wider in more expensive examples), or if you can't get close, an outrageously good (a.k.a. expensive) zoom.
When you crank open the aperture, however, your depth of field gets ridiculously narrow, reducing any margin of error you had on focusing to pretty much zero. Autofocus likely won't cut it at that point, so you have to flip over to manual focus.
Even after that, you'll probably find that everything comes out way underexposed - you could leave the shutter open for a longer time to bring more light in, but that'll exacerbate camera shake and won't freeze your subject, which likely won't be standing still at all.
As for black light, the good news is that most digital cameras are sensitive to the long-wavelength UV light that a black light tube produces, so you can use that as illumination. Additionally, pretty much all lenses will allow those long wavelengths to pass through them (unlike the shorter wavelengths that cause sunburn). Finally, you shouldn't have any problems with any internal filters, etc. in the camera the way you would if you were trying to shoot at the other end of the spectrum (i.e. in infra-red).
So, basically you're looking at an SLR of some description plus lenses. Which one depends on a lot of factors, and is almost a religion with some people.
First things first, what's your approximate budget for the whole kit? After all, there's no point in recommending
L-series glass or something of that ilk if you can't afford it (I know I sure as hell couldn't). Similarly, if your budget is tight, various manufacturers' entry-level digital SLRs have limitations on lenses and capabilities that may well be relevant.