Cassie,
There is an MOD website that advertises teaching jobs at bases around the world. I can't remember what the URL is but try having a look around the internet.... might be something like
www.mod.gov.uk. My daughter was referred to it when she was looking for jobs... like Chris she was unsuccessful!
You could also try obtaining the Cyprus Lion newspaper that is sold on the bases, they have adverts and say who is eligible to apply.