One possible reason could be that Naruto's dad was not around during his childhood. In the Naruto story, Naruto grew up alone and ostracized, and he might feel that his dad abandoned him, which could lead to hatred in a 'dark Naruto' scenario.
The fanfic may develop their relationship by showing how 'dark Naruto' gradually uncovers more and more about his dad's past. With each discovery, his hatred grows. For instance, if he finds out that his dad was involved in some kind of political intrigue within the village that had consequences for Naruto's own life. It could also show 'dark Naruto' trying to break free from his dad's shadow, but in a very negative and angry way.
It could also be due to differences in values. If Dark Naruto had different beliefs about power, friendship, or loyalty compared to his family, and they constantly clashed over these, it might lead to his hatred. For instance, if his family adhered strictly to Konoha's traditional values but he saw them as restrictive.
One possible reason could be feeling neglected. Maybe his family was always preoccupied with other matters, like protecting the village, and didn't give him enough attention. So he grew to resent them.
Another way is through his attitude towards the villagers. He would show no respect or kindness to them. Instead, he would be cold and hostile, perhaps even going as far as to physically intimidate those who cross his path in Konoha.
Often, it portrays Dark Naruto as extremely bitter. He may be shown constantly thinking about Kushina in a negative light, with his every action towards her being full of malice. For example, he might plot revenge against her, even though she is dead in the original story.
It could be that in the fanfic, Team 7 starts to act in a way that goes against Naruto's moral code. For example, if they start using cruel or unethical methods to achieve their goals. Naruto, being the character he is, would not stand for such behavior and this could lead to his hatred. Additionally, if there is a personal loss that Naruto experiences which he blames on Team 7, that would also contribute to his negative feelings.
Perhaps in the fanfic, there is a prophecy that says Naruto's friends will be the cause of his downfall. So, Naruto, fearing for his future and the safety of the village, starts to distance himself from them and eventually hates them. Or maybe his friends accidentally cause the death of someone Naruto loves dearly, which makes him unable to forgive them.
He could hate them because he felt they were absent during his difficult childhood. Naruto faced a lot of loneliness and bullying, and if he thought his parents could have prevented it but didn't, that would breed hatred.
In such a fanfic, Naruto might finally get to understand his family heritage better. His dad could teach him some new jutsu that only the Uzumaki family knows.