It really depends. For relatively short stories I know practically nothing about the characters except what I put on the page -- and that's true for other details as well. It drives some readers crazy that I can't tell them exactly what "The Emmigration" was in my Excavations story, example -- but I didn't need to know it for the story itself, so I didn't bother figuring it out.

There are some things, though, that I do have to know. When I'm doing a story like Counting to Five Thousand, which has a number of characters of different ages who appear in different years, I have to write a chronology of events and births because otherwise the whole thing will fall apart. So I knew exactly when Phillipe, Ned, Goerges-Jacques and Petra were born, exactly when each of the three scenes in the book happened, exactly when Ginevra died, etc. For Returning Were As Tedious I'm keeping a list of the students at Hogwarts at the same time Severus is (not all of them -- only the ones he meets or hears about), including their House and year. But not beyond that. Unless the detail is important to the story in some way, I don't bother.

I can understand JKR, though; especially with backstories. Sirius's dialogue had to be realistic from the word go, and she felt she could accomplish that only if she knew his family background etc. I get that.