You can get as close as you want it to be.
Srsly. If you wanted to dip your nose into the event horizon you actually can.
Like people already posted on this thread, black holes are no more than tiny, really tiny stars with gravitational lensing eye-candy, minus the heat. Scientifically if you were trying to get as close as ~1Ls I would recommend you a serious reconsideration of your plan. Black holes have HUGE gravity, and its tidal force can literally rip your ship apart like a wet tissue paper.
And while there's science, our ships in Elite: Dangerous seem to be perfectly impervious of this effect. You can be up close and personal with any black holes in the galaxy without taking any damage. Just slow down below 1Mm/s speed and fly toward the black hole. Wait until you hit the invisible wall and your ship's computer drops you out of supercruise automatically. When back in normal space, you can get even closer to the event horizon if you wish, but unfortunately your ship will prevent you from trying anything stupid with the dreaded 'body exclusion zone hit' message.
So the final closest distance you can get to a black hole is around 30~70 km depending on the diameter of the event horizon.