Harlan Co. is one of the best places to go off-roading on the planet. Black Mtn. is not really part of the park but it's close to it and there are hundreds if not thousands of miles of trails in the whole area. In fact if you're really into exploring watch out you don't get lost. It is a possibility there. There is a group that leads ATV rides in the area. They get hundreds of riders on some rides. Those rides can be 500 miles long. There is one trail that leads from Bluefield, WV to way west of Black Mtn. in the same mountain range. That trail is said to be 500 miles long itself.
That is about the wildest country in the eastern US. Last time I was there I took a road that runs along the ridge of Pine Mtn. and it took me over 8 hours just to finish that. I think I saw 3 vehicles on that road the entire day. So if you go be prepared. That road / trail runs within a few miles of Black Mtn. and the Harlan park. The place is huge and spectacular. You might want to be aware that a big chunk of that trail runs through a black bear sanctuary. We have those all over where I live though.
The best ATV riding I've found though is actually in WV at the Hatfield McCoy trail at Rockhouse. It's not for the faint of heart. Picture trails just wide enough for an ATV running along a ridge with cliffs on both sides of the trail then picture having to descend a 50 degree slope to get down (unless you chicken out and turn around but you better do it before you get too close to the descent). And that's not one of the worst rated trails. There are lots of places where you can roll 2000 feet down the mountain if you mess up. And it's almost straight down. Other HM trails are much milder but Rockhouse will make you wonder if you really picked the right hobby.
There are some trails the rangers just won't ride if that tells you anything.