homeschooling varies widely, not just from state to state, but from family to family. regardless of the legal requirements, which you can easily research for your book's setting, there is a huge spectrum of what different hs families do. some are very "school at home" in which they "do school" during specific hours, follow strict curriculum, give grades, etc. that is becoming less and less the norm, though.
many homeschoolers are more project and experienced based. they spend a ton of time outside the house, on nature walks, at the science museum, at the homeschool chess/drama/judo/japanese co-op. these kind of homeschoolers don't tend to use grades, or split things up into subjects. it's more of an approach that learning is integrated - baking can be a lesson in reading, fractions, chemistry, and home ec.
colleges LOVE homeschoolers, because they are used to being self-motivated. homeschooling is really gaining in numbers and most colleges are really open to seeing a non-traditional transcript that describes what the homeschooler has been involved in over their teenage years. an internship at the local natural history museum, for example, might take the place of a classroom science and/or history class. also, a lot of homeschoolers start taking community college classes in their early teens and have that kind of transcript to show to universities. here is a good little online article about one homeschooler's experience and admission into MIT:
http://www.suite101.com/content/grown-up-homeschoolers--college-admissions-a211397
there are a lot of different kinds of homeschoolers, and it would be great to be specific about what kind your character is. if you can tell a bit about your character, i could hazard a guess. OR, you could look up some of these types: Classical, Waldorf, Montessori, Unschooling, Charlotte Mason.
and FINALLY, i just recently found out about a new publishing company looking specifically for manuscripts with a main character who homeschools:
http://www.doliferight.com/ i haven't worked with them and don't know them personally, but there it is, if you are interested.
let me know if you have any more questions. (i am a homeschooler to be, though my daughter is only 4.5, so right now i just call what we do living. i have, however, researched the crap out of the whole thing.)