What are black dwarf stars primarily composed of?
The same thing as white dwarfs only colder.
Black dwarfs are theoretically what would remain after a white dwarf star as completely cooled so it does not radiate anymore. The reason it is theoretical is that the oldest white dwarfs are still radiating, and are hot enough to melt steel. It is estimated that we won't see if black dwarfs are real for another 90 billion years or so.
That all being said, based on the theory a black dwarf will consist of iron (final product of fusion which is what will be left when fusion stops at the white dwarf stage) at close to 0 degrees K.