The following are great places to camp, hike, and backpack near and around Naples:
View from Carney Park Rim
Hike the Carney Park rim for an amazing view: as you approach the fork with the sign reading "Admiral Carney Park" (where you turn left before approaching the gate), look to your left to find the trailhead. The trail leads up the southern rim, which has the best views and the best hiking. You can also hike the northern rim, but it is difficult (impossible?) to get to the top, and you have to walk through someone's vineyard/backyard with barking dogs to get down. The NOR Center should have trail maps created by the local Boy Scout troop.
see the Amalfi page for some great hikes along the Amalfi Coast, such as the Sentiero degli Dei (Walk of the Gods)
see the Gulf of Naples Islands page for hikes on these islands, including up Mount Epomeo on Ischia and to the Arco Naturale on Capri