Camping in NYC? I don't know of any, but you might find something on Long Island or in Westchester County. How about googling Bear Mountain - I think thats a state park on the Hudson River a bit north of NYC.
On the other hand, the homeless have been doing it for years........
And if you are willing to camp as far away as Philly, you might try a bit farther up the Hudson. Say around the Shawgunks (mountains that are famous among climbers maybe an hour and a half north of the city). Google NY State parks and see what comes up.
I just did that and found a lot of options. There is more camping there than I thought. You might also google Camping near New York City. A lot of things come up. When you settle on something, PM me. I was born in the Bronx, and grew up in Tuckahoe (near Scarsdale and White Plains). I've been in Cleveland since graduating high school but I have some knowledge of the area. If I can give you any info that is not on the websites you pick, I'll be glad to help.
One suggestion. Ride your bike to a Metro North train station that has daily parking and take the train into NYC. Traffic in the city is unbelieveable and parking is expensive at best. You will blow quite a bit of money on a daily basis - food can be expensive (and bathrooms hard to find). I love NYC and have walked and walked there for hours on end. Plan, plan, plan, and/or just wing it. Its a wonderful place.