An essay that considers the effect of the fictional character on our real culture clearly isn't fan fic. I'm not sure about the case where the effect on the culture of "the world" that the character is in is considered.
For a bio, I don't know if there's some distinction about to what extent fictional events are used in the bio. I.e., is the bio just a rehash of the original stories.
Thanks in advance.
--WouldBe
If your bio is simply going to draw from established work, then it's not fanfic at all. If you are going to make up additional material then that's a different matter entirely.
If you summarise the contents of the Harry Potter series, for example, into a "biography" of HP, then it's just an article or scholarly essay. If you added events that J K Rowling never mentioned, then it's fiction, and thus pretty much by definition fan-fic.
Can I ask what the purpose of such a biography would be? Who would be its intended readership? How would you consider publishing it? These questions have a bearing, to some extent, on whether material is considered "fanfic" or not (and whether it breaches copyright or not, which is probably a more serious point of consideration).
On the other hand...there have been a number of credible biographies of fictional characters---the late SF writer Philip Jose Farmer wrote two that I know of, on Tarzan and on Doc Savage---and something like that might stand a chance of being professionally published. (Copyright disputes disregarded for the moment.)
(Do the professionally-published Star Trek novels and such count as "fan fiction"?)
The work might include critiques of other writings about the character, real or imagined. It would be written without mentioning the licensed work (novel, movie) itself. There might be a fictional other biographer that is the foil to this biography. There would be, perhaps, new fictional events in the real world. Sounds odd, but I thought it might be interesting to fans. It would be written unblinkingly like a serious biography, so the dry humor would have to work or it would totally flop.
BTW: The writers who write works for licensed worlds/characters, etc. with the permission of the license holders are "tie-in" writers. They have an organization: http://www.iamtw.org/
If that is the case, then you need to be aware you can't expect to make any money from it, and if you choose the wrong subject, you could potentially find yourself the subject of legal action (cf. the row over the Harry Potter encyclopedia).
That's not remotely to say you shouldn't do it if it's something you really want to do.