There are only a couple of acceptable parks in/near Asheville. Campfire Lodgings is full hookup and has very good reviews. Lake Powhatan Recreational area is another. It has full hookup too but is a bit more of an effort to get to. Mama Gerties is another one. Very good reviews and has all you need. Bear Creek is good and is the nearest to the Biltmore Estate but it sits right at the interchange of I-40 and I-26. Have stayed there and it is a very nice park but road noise can be an issue depending on where your site is. These parks are the only ones I would recommend that are IN Asheville.
Keeping an open mind don't rule out Maggie Valley. It's only 30 miles out of Asheville and the parks there are quite nice. There is the KOA Asheville West park that is 13 miles west of Asheville. Stayed there before too and found it acceptable. Sites are a tad close for my liking but it served our needs for that trip to Asheville.
We are season pass holders to the Biltmore so we go occasionally as it is only 70 miles away from home. Asheville crimes rates are going up so don't be downtown at night. Homeless are all over the city now. Just be aware.