If the Earth became a black hole (which it couldn't very easily; there isn't enough mass in the earth to make one that would be stable) there would be no affect on the orbit of the moon at all. The only difference would be that the rotational centre of the Earth and the Moon - the point where the gravity of the Earth and the Moon pull equally, around which they rotate - would be a few thousand miles above the black hole in the direction of the moon, rather than a few thousand miles underground as it is at present.
