She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
Alexis could also be a bit frustrated. She might feel like she's caught in the middle of Castle and Beckett's miscommunication. And she may try to stay out of it at first, but as it continues, she has to step in to clear things up.
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.
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.
Kate might get defensive right away. She could say things like 'How could you think that of me?' and be angry at Alexis for not trusting her. This is a common reaction when someone is wrongly accused in a fanfic or in real life.
There are definitely some great 'castle beckett and alexis fanfic' pieces. If you like stories that explore the emotional side, you might find some where Alexis has to deal with the dangerous aspects of her father's and Beckett's job. Others could be more light - hearted, showing their day - to - day interactions in a humorous way.
He might be in denial at first. He could keep trying to convince himself that there's no way Beckett would leave him.
Fanfiction.net is a great place to start. It has a large collection of fan - created stories, including ones related to Castle, Beckett, and Alexis.
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.
She is probably angry and frustrated. She would demand Castle to explain why he thinks so and would be hurt that he doesn't trust her.
Castle would probably be shocked at first. He'd be thinking 'Why is my daughter doing this?'. Then he'd start trying to talk to her, using his charm and wit to figure out what's going on.
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.