Best time is any time but the 'season' which approximately runs from December to late March or early April depending on how cold the northern tier of states happens to be.
You don't want to camp directly in Key West, I'm not even sure if there is a CG there. Marathon Key or further south are the places to look. Also I doubt you find any camp sites with less than FHU. The Keys are small geographically and if anyone puts up a CG it will be FHU considering land prices.
The prime spot is usually considered to be Bahia Honda State Park:
Link I think you can make reservations a year or so in advance but check
the reservations page. You will want to try to get reservations the day that they open up, it is a very popular park. All the Keys state parks with campgrounds are good if Bahia Honda is not available so check those out.
We usually camp at marathon but that may be to far away from Key West for you. One thing, if you want to snorkel or SCUBA then you need a period with lower winds. High winds mean turbid water and no visibility.