Depends on where you live in NJ, of course, but there are plenty of great backpacking areas in PA, as well as upstate NY (in the Adirondacks).

The Susquahannock State Forest in PA is only about 3-4 hours away from Trenton and provide lots of backpacking trails and such, for one example.

The George Washington and Jefferson National Forest in VA, WV is about a 5-6 hour drive from Trenton and also offer lots of backpacking trails.

And Harriman, in NY, is also a good choice for something closer in, as BZH said.