He might try to prove himself. Naruto has always been a determined character. So he would likely strive to show his family that they are wrong about him. He could do this by achieving great things in the ninja world, like mastering new jutsus or completing difficult missions.
In fanfiction where Naruto is hated by his family, it could be a very dark and angsty take. Maybe the author wants to explore the idea of Naruto facing even more hardships on top of being an outcast in the village. It could lead to a very different Naruto, one who is perhaps more bitter or desperate for acceptance outside of his family unit.
Naruto could try to communicate. He could sit down with his family members and try to understand why they hate him and also explain his side of the story.
Naruto could develop a very different personality. Instead of his usual cheerful and determined self, he could become withdrawn and angry. This new personality would affect his relationships with others in the village, and might even make some people fear him more than they already did.
Another reason might be related to family politics. In some complex family structures in the Naruto universe, there could be different factions. Naruto might be on the wrong side of the power struggle within the family. His views or actions could be seen as a threat to the traditional power structure of the family, so they hate him.
In a modern fanfic, if Naruto is hated by his family, it could change his entire life trajectory. He may be forced to leave his home earlier than in the original story. This would expose him to more danger at a younger age. His relationships with other characters would also be affected. For instance, his relationship with his teachers might become more complicated as they try to navigate the situation. Additionally, Naruto's view of the world could become more cynical, but he could also strive to create a new family of sorts among his friends, which would be a significant aspect of his character development.
Maybe in that fanfiction, Naruto's over - eagerness to be recognized could be misconstrued. His loud and boisterous nature might annoy people. For example, if he constantly interrupted others or pushed his ideas too hard without considering others' feelings, it could lead to dislike.
Naruto might also be determined to change his brother's mind. He would keep showing his good side, like his kindness and his strong will to protect the village. He could think that by being a better person, his brother will eventually stop hating him. He may even go through some tough training to prove himself not only to his brother but also to the whole village. And in the process, he may face a lot of difficulties and rejections from his brother, but he won't give up easily.
It's not overly common, but it does exist. A lot of Naruto fanfiction tends to focus on other aspects like his relationships with his friends or his growth as a ninja. However, the concept of his parents hating him can offer a unique and often darker take on the character. Writers may use this to show a more jaded Naruto, one who has to overcome not just the hatred of the village but also from his own family.
In this fanfiction scenario, Naruto's reaction could be complex. Given his character, he would likely be deeply affected by his parents' hatred. He might have dreams where he tries to seek their love but is constantly rejected. As a result, he may focus all his energy on his ninja training, not just to prove himself to others, but also in the hope that one day his parents will see his worth. He may also develop some trust issues, being cautious around those who show kindness, always suspecting an ulterior motive behind it. But as he grows and experiences more, he could gradually come to terms with the situation and find his own path in life.