How do black holes relate to Newton's law of universal gravitation?
Newtonian gravitation applies to black holes at a distance.
At a distance a black hole is just another body and Newton's laws of gravity apply. Objects will feel the gravitational attraction of a black hole and objects can orbit around the black hole.
Closer to a black hole Newton's laws of gravity break down as relativistic effects come into play.
Objects orbiting closer to the black hole will be travelling fast enough that relativistic effects such as time dilation will be experienced.
Objects getting close to the black hole will experience the effects of General Relativity. Time will slow down and gravitational tidal effects will occur.