A character cannot be, of itself, copyright. I can write stories about anyone I chose, in complete confidence of the long standing principle that there is no copyright in ideas. As the saying goes - the devil is in the details. Television, comic books, anything illustrated will strengthen the copyright owner's case - the fully visual depiction is the strongest protection. On the other hand, Sherlock Holmes, instantly recognisable even in profile - meerschaum pipe, deerstalker cap, angular features should be approached with extreme caution.
Most fans agree that as long as no profit is made, then no one is going to care what they write.
There are circumstances where this is not the case however. Erotica can be a tricky problem for instance. More than one fan fiction author has found a terse note - or even a Cease and desist order, making a sudden appearance, when a piece of fiction has somehow worked its way through to the copyright owners. Often it is seen as inappropriate to the image of the original, or potentially embarrassing for the actor portraying the character. Some people are genuinely disturbed by the fantasies their character has generated.