Odds of Getting In
Your chances of getting the job are fairly decent, provided you can offer good references, demonstrate a positive character, and show that you are skilled at working with children.
Possessing certain skills may also help you get hired. As mentioned, camps like counselors who can supervise in-demand activities. If all you can teach are sing-along songs, you're not going to be as appealing as someone capable of teaching advanced archery.
Counselors sometimes work at the same camps that they attended when they were children. This is by no means a requirement, but it can't hurt if the senior staff already knows you. Actually check that—it might hurt if you were a constant terror as a camper.
Overall your chances of getting the job are decent. It just depends on how many applicants the camp receives for the summer. A good strategy is to apply to a wide number of camps in your area. Odds are that at least one will bite, and then you're set for a summer full of finger-painting and food-fights.