Perhaps in the fanfiction, Harry avoids his family. He doesn't want to be around them because they are always criticizing his relationship choices. For example, if he's in love with a Muggle - born witch, and his family is against it in the fanfiction, he would start to avoid family gatherings and show his hatred through his absence.
Often through his nightmares. In fanfiction, he might wake up in a cold sweat, reliving the moment when Voldemort killed his parents. His reactions to things related to his traumas are also a sign. For instance, if he sees a Dark Mark - like symbol, he could freeze or have an anxiety attack.
It wasn't really shown directly in the movies. But we can assume that if he knew about it, he might have shown his dislike through his actions towards those who misrepresented the Jedi teachings which fanfiction might do.
His thoughts are also used to portray his hatred. The author might write from Harry's perspective, and we can see his negative thoughts about people. He could be constantly thinking about how everyone is out for themselves or how no one really understands his pain, which then colors his view of everyone around him.
He might openly oppose their policies. For example, if the Ministry tries to pass a law that restricts the use of certain spells for self - defense, Harry could publicly speak out against it, giving interviews or writing articles in wizarding newspapers.
It could be shown through his avoidance. He might always try to stay away from Annabeth in the hallways or during group activities.
One possible reason could be jealousy. Maybe the sister is getting more attention from their parents or at Hogwarts. For example, if she is a star student and Harry feels overshadowed.
Well, it could be that in love fanfiction, the authors are trying to create more drama. They might make Harry's family oppose his romantic interests. For example, if Harry falls in love with someone from a different 'group' in the wizarding world, his family, as written in the fanfiction, could be really against it. This would lead to Harry hating them because they are getting in the way of his love.
It could lead to some fanfiction writers feeling discouraged. If they know that the main character they are writing about hates their work, they might lose motivation. For example, some might stop writing stories related to Harry Potter.
Often, it's shown through his internal thoughts. The writer might describe Harry having angry and bitter thoughts about his parents when he's alone, like constantly thinking about how they left him to face so many difficulties. For example, he may think 'My parents left me to deal with Voldemort, they didn't care.'
It could also lead to new alliances for Harry. Since he has refused his family, he might seek support from his friends more, strengthening those relationships. This can change the power dynamics in the fanfiction. For example, Hermione and Ron might become even more important to him, and together they could face new challenges that were previously unthinkable without his family's influence.