Camping in NYC sounds like a bit of an impossibility, but while there isn't any camping in Manhattan, you don't have to go far for a nice overnight trip-even over in Brooklyn, there are clean, well-patrolled options that will convince you that the city and its problems are far, far away.
The closet camping option is over at Jamaica Bay where there are plenty of campsites and tons to do around them, primarily water sports and beach hangouts. If you're a casual cyclist, you'll love the Jamaica Bay Greenway, which you can use to get around, and there are tons of running trails and roads in the area. The campsites are clean and easy to access, and there are almost always park rangers in the area.
If you want to get a bit farther out of the city, in the summer, take the ferry or drive to Sandy Hook for a weekend and camp at Horseshoe Cove. The beach is great, and the peninsula offers a great chance for casual road riding (you can easily go into New Jersey for nice roads), mountain biking is close by at Hartshorne, and the running, road or trail, can't be beat. A word to the wise: There is a nude beach at Sandy Hook, so if you're on a family trip and don't want to expose the kids to New Yorkers in the buff, make sure you steer clear of the clothing-optional section.
Lastly, if you want more traditional in-the-woods rustic camping, Harriman State Park (about an hour north of the city) has you covered with options for tent and cabin camping, tons of hiking trails and lakes, and right over at adjacent Bear Mountain, there are plenty of kid-friendly activities year-round.