One of the best hikes or strenuous runs in the area is to climb Mt. Mitchell from Black Mountain Camp Ground. This is a full day hike or run and is quite strenuous. It's basically 5.5 miles straight up, and hikers or runners need to be experienced, fit, and well-prepared with water and fuel. Get on the trail early in the day.
The Trail is wooded and gains 3600' in elevation over 5.5 miles to reach the summit. Of note, Mount Mitchell is the highest peak east of the Mississippi River, so this is definitely a good notch in a hiker's belt. Hikers will enjoy varied forest that changes with the elevation gain. Some is old-growth forest, and there is a gorgeous alpine meadow, which, in the Spring, is dappled with wildflowers. Of course, the main goal of this hike is reaching the top, and the expansive panoramas are well worth the effort. This is truly one of the most beautiful views out there.
If you're looking to turn your hike into an overnight trip, the small primitive campground in the state park has nine tent pads, each with a picnic table and grill. Sites are available on a first-come, first served basis. There's a restaurant, camp store, and a comfort station with flush toilets, but no showers. Camp store employees can also direct you to a few back-country camp sites if you're looking to get away from tourists and wake up in the morning to an incredible view of mist-covered peaks. Totally worth it!