What are some good backcountry camping sites near New York City?

You're in luck, my friend, because there are literally hundreds of thousands of acres of backcountry around NYC. Not really backcountry, there's camping inside the city limits. In Brooklyn, actually. Floyd Bennett field is an airport-turned-park. It offers finished campsites and RV hookups, surf fishing, biking, classes, etc. It's also a favorite place for RC enthusiasts. $20/night. If you've got a car, drive northeast on the NYS Thruway and you'll be at Harriman State Park and Sterling Forest within an hour. Both offer challenging hikes. A bit further, Bear Mountain. Then there's the Shawangunk Mountains -- a veritable climber's playground. The Catskills, about two hours north, offers some serious challenges. The Devil's Path is considered one of North America's most challenging trails. Further north, the Adirondacks offer some of the most remote places in the northeast. If you have to rely on mass transit, don't despair -- you can get on the Port Jervis line of New Jersey Transit. There are four consecutive stops; Suffern, Tuxedo, Sloatsburg, and Harriman; that leave you at nearly a dozen or so trailheads. By train, look at 2-2.5 hours to get to any of them. Taking the train has some different challenges associated. Some trails are a mile or so from the stop. Take that distance into account as you plan. If you're taking the subway to Penn Station, (where you pick up NJT,) avoid times when the subways are crowded. It's already a PITA with a pack and trekking poles, but without a seat I'm standing for at least an hour before Penn St. As for camping, there are some designated campgrounds in the state parks. I don't use them, opting to find my own. Now, I'm indebted to these places for making me whole, and I respect them deeply. I camp 100' from a trail, and not where ground meets water. Pack it in, pack it out. No glass. Piss 50' from water, dig your cathole 6" deep, and save the plug to put back. I am a member of, and have volunteered for, the http://www.nynjtc.org/, a group responsible for building and maintaining thousands of miles of trails in the area. They have the best maps on Earth -- they made the trails they're printing them for. They're coming out with revised editions of them now. The maps are printed of Tyvek -- it doesn't absorb water, and I challenge you to try and tear it. It's the stuff FedEx envelopes are made of. You will have a map, right? Right? You should give them money, too. I'm telling this to you with confidence that you'll use the information wisely. If I find out you haven't, I'll regret sharing it with you. I hate regret.

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.  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. 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.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.  Get this map pack and the possibilities are nearly endless http://nynjtc.org/product/new-catskill-trails-map-set-maps-hiking-catskills

The Appalachian Trail passes maybe 50 km North West of New York City.  It is a great trail for hiking and camping, although maybe a not quite "back-country" especially along that section.  When I was living in Washington D.C. I used to take the commuter train out to Brunswick or Harper's Ferry to go hiking on the AT.  These are still some of my fondest memories.