If you're going to go hiking in the Black Hills, you have to hike up Harney Peak. The Black Elk Wilderness has some nice hiking trails, but I'm not sure there is more than one night of backpacking in terms of length, but it is a gorgeous area.
I've gone backpacking at both Badlands and Wind Cave NP, which is just south of the Black Hills. The Badlands has one primitive campground that is free, but you can also park here and hike into the wilderness area and camp anywhere (I think you have to get a free permit though). Wind Cave has areas open to backpacking and camping. They have their backcounty split into zones, and you just tell them on your permit which zone you plan on spending the night in. Wind Cave is sort of a foothills topography where it goes from open prairie to forested hills. You could also go hiking in Custer State Park. I've camped in the French Creek Natural Area and that was a nice hike along a stream.
The Crazy Horse Memorial is a nice place with a lot of information. I don't know when in the summer you're driving through, but during the first weekend of June, you can hike up to the top of the monument in progress (up to just below the head).
I had a temporary job with Black Hills National Forest for six months, so if you have any other questions, I'll try and check this occasionally to reply.