I tend to go through phases of reading fan fiction - Since I tend to focus on characters more than plots, I like the stories that give different or more in depth looks at my favourite characters. A lot of the time the person I would like might be a supporting character rather than the main hero/heroine so I like to learn more about them even though it is the writer's imagination but if it is written well, you can immerse yourself in it.
I write a lot in general, but not really anything I would call fanfiction. I tend to be drawn to universes and worlds rather than characters, a trend which doesn't seem to fit in with most fan fiction sites. I've read a fair bit but generally so much has been of low quality that I haven't spent the time to find the gold (though occasional lucky finds still stand out in my memory as great pieces of fiction).