I want to take facial portraits, indoors, with with a black background?

I have a Nikon d3100 but no other flash. Have you any advice on how this can be done?

A camera flash is really harsh so put a piece of rice paper over it. Or have a day light lamp over the subject, over for young and under for old.

You can't really do a good portrait with a simple flash on camera - so, don't use it. Use whatever lights you have - either incandescent or fluorescent - to provide standard "portrait" lighting. I know a quick search on-line will tell you roughly where to position the lights so I'm not going to bother to explain.

Black background could be a challenge but make sure the subject is far enough away from the background that there are no shadows.

That should do the trick. Just be ready to do lots of experimenting, some colour adjustments. And retouching.

Have fun!

It depends how you want to light it (low key, high key).

If you want a single light source at an angle with no fill you are going to need to use a continuous (ambient) light source only (not your on camera flash).

Position the light close to the subject.adjust angle to subject for required lighting effect. Have the subject far enough away from any other reflective surfaces. Spot meter on the lit side of the face.

You can use flash, but you will end up with frontal (flat) light which will not define the subject as well. IF you want to do this, then I suggest using flash on manual (not TTL) and shooting at or close to the max sync speed (around 1/200 sec) with an aperture of around f8 or so. This will kill the ambient light and you light the subject subtly with the flash. You will not get a shadow/highlight portrait with this method, but the b/g will be black. Again make sure there are no reflective surfaces behind the subject.