I don't know of many campsites round there, allthough think theres a youth hostel very nearby? If you drive 20 mins north though into the white peak around Illam way there are tonnes (and possibly better area for running). Its a somewhat quite corner of the Peak which is all part of the charm. Alton has lots of pubs of varying quality for the Alton towers crowd.
That's right, I'd camp at one of the many small campsites close to the Manifold valley. Hulme End is a lovely spot with several campsites close to the river and the Manifold Hotel (pub). Google Hulme End camping. Lovely spot. There are several campsites closer to the Churnet but several of them look more like caravan sites in the main. Another plus of camping in the Manifold is the easy access to some great gritstone bouldering - Roaches, Newstones - Boldstones. The Churnet is a bit damp at present (I was there yesterday) due to the tree cover.