Will a diamond conduct electricity and, if so, how good a conductor is it?

No, diamond is an insulator.

There is such thing they are DVD diamonds,synethic diamonds at that. You might want to look into that for more precise info. Just like ruby's are used for the infared light. The dvd diamond can transmit the information though green light which is a faster component. This is more for computer technology. Diamonds give off heat. And if they are sysnethic there cut in different angles to trasmit information through the green light which is faster than the infared beam. New technology.
I did a report on this in high school and proved my physics teacher wrong that diamonds do in fact give off heat. (keep in mind synethic diamonds) there calling dvd diamonds.