1) Harriman State Park:
Harriman State Park is a good place to start. Officially you are required to camp at one of the shelters. I spoke to a ranger once and he said that as long as you’re camping within sight of the shelter you are good. Unfortunately on weekends you can sometimes encounter an entire Boy Scout troop camped about a shelter. You are better off going mid week or in the fall.
Despite the “official” policy, if you get yourself far enough off trail you can find back country places to camp and nobody is going to bother you.
Map link: https://goo.gl/maps/nywztQY5wbeUMesR9
2) Round Valley Recreation Area:
Round Valley Recreation Area in NJ is quite nice. It is a state campground but the sites are only accessible by boat or a 3 to 6 mile hike (depending on which site you get). As with most state park campgrounds, you need to reserve your site (there is a reasonable fee). The boat-in sites next to the water are often booked well in advance on summer weekends but the hike in sites, which are a little away from the water, are always available. If you are lucky, you can get a water site even if you hike in. The sites are very basic, fire ring only. A few have picnic tables, but most don’t.There are potable water and outhouses available with a short walk. The sites are nicely spaced and provide a decent backcountry feel.
Map link: https://goo.gl/maps/yJGZzidXReLrpZdy7
3) The Appalachian Trail in NJ and NY:
In some of the more popular areas in NJ (such as around Sunfish Pond) you are supposed to camp at one of the designated backcountry sites or shelters, but you won’t be alone. It is easy enough though to keep hiking past the designated sites and shelters and find a nice spot off the trail. The sections of trail in Stokes State Forest and the Delaware Water Gap National Recreation Area offer some excellent back country camping if you wander off trail a few hundred feet.
Map link: https://goo.gl/maps/4YxzAsXEbX658Dfe8
4) The Catskills:
This is a backpackers paradise. For the most part you can camp anywhere on state land as long as you are at least 150 feet from a trail, road or water and below 3,000 feet in elevation. There are a few shelters too, if you are into that thing. There are some designated sites that are within the 150 ft of water/trail/road rule, but they are first come-first serve. Some awesome trails to consider are Devils Path, Overlook Mountain Trail, the Escarpment Trail, The Long Path, Black Dome Trail, Warner Creek Trail, Wittenberg-Cornell-Slide Trail, Peekamoose-Table Mountain Trail, Giant Ledge-Panther Mountain Trail, Dry Brook Ridge Trail and the Trout Pond and Mud Pond trails.
Map link: https://goo.gl/maps/uhAHmSK2beU9F8xAA