When we moved into our apartment in May, there was a stray cat that would lurk right outside our door. Our neighbors told us that the people that lived here before us owned this cat, but just left it when they moved!

It is ALWAYS outside our door, so we started feeding it. It used to be VERY skittish, but since we've been feeding it (and I call it an "it" bc I have no idea whether it's a boy or girl) it has become very friendly. Everytime I walk out my door it runs up and rubs it's body up against my legs

It even likes Hobbs & Hootie! It will just lay there and let them sniff it.
So, my question is, what should I do with it? I am a little hesistant to let my dogs too close to it because I'm sure it hasn't had any shots lately. I really want to take it to the vet to get shots, flea & heartworm prevention, and really call it our own, but I KNOW it will not let me pick it up and take it anywhere. My biggest fear...what will happen to it when we move out?!
So, what do you think? What should I do? I really want to help it and give it a good life and keep it, but how?