Maybe Alexis has a different vision of the original story or characters. Fanfic often takes creative liberties that might not align with Alexis's view of Beckett.
It could be that Alexis has a strong attachment to the original canon regarding Beckett. Fanfic can sometimes deviate too much from what Alexis considers the essence of Beckett, thus leading to dislike. For example, if the fanfic changes the personality of Beckett in a way that Alexis doesn't like, it can create this hatred.
Perhaps Alexis has had a negative experience with some Beckett fanfic in the past. Maybe the writing quality was poor, or it contained elements that Alexis found offensive or just not in line with what she/he expected from Beckett - related stories.
In the fanfic, it could be that there are signs that Beckett is hiding something. Perhaps he has unexplained absences. He might say he's at work but Alexis can't reach him during those times. Also, he could be overly protective of his phone or personal space, which makes Alexis suspect that he's cheating.
Alexis could also try to engage in discussions with other fans of Beckett. By sharing her/his views, Alexis might gain new perspectives and either change her/his mind about fanfic in general or at least find a subset of fanfic that she/he can tolerate.
It could be that Castle is overprotective. He loves Alexis so much that even the slightest sign of Beckett not being on the same page as Alexis in Castle's eyes makes him think Beckett hurt her. Maybe Beckett gave Alexis some career advice that Castle thought was too pushy.
It could also be due to the way the fanfic writers handle their characters. If they write Bulma as a prideful character and Chichi as someone who challenges that pride, either directly or indirectly, Bulma might be written as hating Chichi. Fanfic often takes creative liberties and this could be one of the results.
Perhaps the fanfic often involves modern and sometimes out - of - character interpretations. Apollo, as a figure deeply rooted in ancient mythology, might be seen as 'hating' these fanfics because they deviate too much from the well - established stories and personalities. The fanfic could give Artemis new traits or put her in situations that Apollo, in the context of the original mythos, would not approve of.
Maybe Naruto has some personality clashes with the Hyuuga characters in the fanfic. For example, the Hyuuga clan is known for their traditional and sometimes rigid ways, which might go against Naruto's more free - spirited nature.
Well, without having read the actual 'beckett and alexis killed fanfic', it's difficult to give a detailed account. However, generally speaking, it might be a piece of fan - made fiction that focuses on a plot where Beckett and Alexis are somehow related to a killing. Maybe they are the ones who do the killing, or perhaps they are witnesses or victims in a more complex scenario.
Alexis might confront Beckett directly. She could say something like 'I think you're cheating on me. What's going on?' This is a very straightforward approach.
Perhaps there have been some rumors or whispers in their fictional world that have reached Alexis' ears. Even if they are baseless, they can plant the seed of doubt in Alexis' mind, making her think that Kate is cheating.
Common themes in fanfic usually revolve around relationships. For Alexis Hurts Beckett fanfic, it could be a complex relationship filled with misunderstandings that they have to overcome. There might also be themes related to power dynamics if one character has more power or influence than the other. Another possible theme could be self - discovery, as they interact with each other and learn more about themselves in the process.