In my experience, it is difficult to find a good fanfic with an asexual character. Usually the fic either reads like a “very special episode” on asexuality, or just generally portrays asexuality horribly.
One of my major pet peeves with fanfics featuring asexual characters is that the entire fanfic ends up being entirely about the character’s asexuality. There’s no other plot, motive, or storyline; it’s just about “dealing” with asexuality. What I mean by this is that usually the asexual character is in a relationship, and a large majority of the fanfic deals with the sexual partner dealing with and trying to understand their asexual partner. And while a few plots like that are all well and good, when it’s overused it starts to become very similar to TV shows like Glee and many others that make homosexuality a character’s key personality trait and their entire plotline.
Another thing that sometimes bothers me is the lack of fanfic with a female asexual character. To be fair, the lack of female characters is a problem in our media in general and thus is naturally a problem in fanfiction.
So for my Asexy April fanfic I chose a fic with a female asexual character where the story doesn’t focus entirely on her sexuality: Speak until the dust settles in the place light refused to go, an excellent little Teen Wolf fic featuring an asexual Kira!