I don't usually write OC fanfiction. I only have one story that's filled with OCs, and that takes place after the days of Harry Potter, and focuses on a different character's life at Hogwarts.
With me there are two different kinds of fanfics. There are the ones that focus on JK Rowling's characters from the books, their relationships, their personalities, continuations of the original sory, etc. Then there are OC stories, that revolve around an original character, preferably without the presence of too many major characters from the original books.
The OC fic I'm writing is about a character loosely based on myself, which I have placed in the Potterverse and tried not to make into too much of a Mary-Sue, and it takes place after the Harry Potter septomology (shut up, it's a word!) is over. Although I might bring in some of the major characters of Harry Potter later on, they will have minor roles in my story.
Those are my two scents.