Don't try camping in the keys until about Nov unless you are a committed blood donor! (mosquitoes).
There is a nice overnight trip to Carl Ross Key (25° 2'35.72"N, 81° 1'5.95"W). You will need to reserve the campsite with the rangers at Everglades National Park. Carl Ross is ~9 mile paddle...put in at Flamingo, Everglades Natl Pk. You might share a camp site with someone who motors in, but the camp sites (about 2-3) are pretty far apart.
There is only one other island campsite in the keys, Nest Key (25° 8'58.64"N, 80°30'50.89"W). Put in at Key Largo.
However, there are some nice beach camping sites in the Keys, for example in Everglades Nat'l park. You can only reach these sites by paddling (or motor craft), thereby excluding car campers. Again, you need to reserve the campsites with the rangers. Telephone Everglades Natl Pk for details.
There is beach camping all along the keys (State Parks and other specific sites, only) but beware: these can be crowded with drive-in campers.
In addition, check out http://www.dep.state.fl.us/gwt/paddling/Segments/Segment%20Home.htm
for a description of the Florida Circumnavigational Saltwater Paddling Trail.
Last, you could drive all the way to Key West and take a 2 hour ferry ride (with your kayak on board) to the Dry Tortugas National Park and camp there 1-2 nights. Google the Dry Tortugas for descriptions, pictures, info.