Interesting question. I have to say, as others have, you must definitely check the laws for your state. Each state is different, so any specific information you get on here that says "oh, just do this" may not be applicable to you in your state.
As far as the homeschools in churches, my church has two.
The one I am involved in has close to 130 families and it is a teaching co-op. In order for me to be involved, I have to be involved by either teaching, setting up, cleaning up or doing administrative work.
The second is a cross between a private school and a homeschool. The children meet a couple of days a week for instruction and work at home the rest of the time. In our co-op we have high school credit classes that work on the same principal, our students show up for tutoring, then do all their work at home.
So, that's just a 411 on "church homeschools" according to my personal experience, there may be other types, but I have no experience in them.
My advice to you, since you asked it to seriously consider why you want to homeschool your child in the first place. I served as Administrative Director of a homeschool group for two years, and during that time, I counseled many many families, and a red flag for me was when the parents said, "I don't have time to do this"
so that's why I say, to make sure you want to do this. I love homeschooling my kids, and I'm a proponent, but not to the degree of being able to say it is for everyone all the time. If you aren't happy with the level of education, then a tutor to enhance your child's public education may work for you.
If your child is having serious problems and the public school environment is actually detrimental (like your child not being able to resist negative peer pressure, drugs, things like this)
then I would suggest that this would be a priority to you and you should find the time to make sure your homeschooling experience is a success.
I'm not trying to bash you over the head, I'm just giving you the same advice I've given others that came to me with the same circumstances.