What does a black dwarf look like? A black dwarf is a hypothetical object that is the end result of the cooling of a white dwarf. None yet exist, because there hasn't been enough time in the age of the universe for them to cool down sufficiently. The oldest known white dwarfs are at ~3800 K and would glow orange. It will take many billions of years before any are cool enough (~750 K I believe) to not emit light visible to the naked eye.

So, what would a black dwarf look like? I'm assuming it is illuminated by some other source of light such as a star. 

The question could be rephrased: What is the visible absorption (and emission) spectrum of electron degenerate matter?

This is my attempt at an answer: Probably not black. A lot would depend on whether electron-degenerate matter behaves as metallic or dielectric, based on its band-structure. My guess would be metallic, considering the electrons are dissociated from their parent nuclei in electron-degenerate matter.

Because the surface gravity of a black dwarf would be so strong, I'd expect the surface to be very smooth and relatively free of Lambertian scattering. 

From these two features, I'd make a tentative prediction that a black dwarf would appear as a mirror-like sphere.
The material at the surface of a white dwarf is not degenerate. 
The "visible" surface is defined as where the optical depth exceeds some threshold and this will occur at a low enough density that even at a few hundred kelvin, the ratio of the Fermi energy to the thermal energy is too low for significant degeneracy.

In addition, at these temperatures, the electrons attached to atomic hydrogen and helium would not be ionised and the material would mostly be in atomic and molecular states.

In terms of appearance a reasonable comparison would be with that of the very cool T- and Y-dwarfs; although some of the more complex chemistry that occurs in those atmospheres would not occur in very cool white dwarfs because the trace quantities of heavier elements would probably have sunk out of the atmosphere.

Edit: I note that there probably are white dwarfs cooler than 3000K. This is the upper limit to the temperature of an invisible companion of to the pulsar PSR 2222-0137. Kaplan et al. (2014) say that this most likely a massive ($\sim 1 M_{\odot}$) old white dwarf that has cooled so much that they cannot detect it at visible wavelengths - so probably you would call this a "black dwarf", though the term is not used in the scientific literature.