In this type of fanfiction, it's possible that the story delves into Naruto's backstory and how being Sarutobi's son would have affected his childhood. Maybe he would have grown up with more guidance and support, which could have changed his personality. Instead of being the lonely outcast that he was in the original story, he could have been more confident from the start. The fanfiction might also introduce new characters related to Sarutobi that play important roles in Naruto's life. There could be new adventures and challenges that are unique to this new family dynamic, such as Naruto having to uphold the Sarutobi family name and legacy within the Hidden Leaf Village.
In a 'naruto son of asuma fanfiction', perhaps the story focuses on how Naruto's relationship with Asuma affects his relationships with other characters. For example, his bond with Shikamaru could be even deeper since Asuma was Shikamaru's mentor. And Naruto might inherit some of Asuma's fighting styles and philosophies, which would change the way he approaches battles and his ninja way.
Obviously, Naruto would be one of the main characters. And since it's about him being the son of Sarutobi, Sarutobi himself would be a crucial character. There would also likely be the usual Naruto characters like Sasuke, Sakura, Kakashi etc. as they would still be part of Naruto's life in the fanfiction, even with this new family relationship.
It could be Orochimaru. Since Orochimaru is always after powerful ninja and their abilities, if Naruto is now the son of Asuma, he might see Naruto as a new target with unique potential.
Well, a plot twist could be that since Naruto is Hiruzen's son, the Akatsuki targets him differently. Instead of going after the Nine - Tails inside him in the usual way, they might try to use his relationship with Hiruzen against him. They could try to blackmail Hiruzen into making certain concessions for the safety of his son. Also, Naruto's training might be more intense and different. He might be taught more about the political side of being a ninja by Hiruzen himself, which changes his perspective on the ninja world.
One of the popular characters paired with Sarutobi is Jiraiya. They are both important figures in the Hidden Leaf and their relationship, whether it's as comrades or as teacher - student in some fanfic interpretations, is always interesting to explore.
Obviously Naruto would be one of the main characters interacting with him. Naruto would look up to Sarutobi as a mentor and a source of knowledge. Their interactions could range from training sessions to deep conversations about the future of the village.
If Naruto were Hiruzen's son in a fanfiction, his character development would take a whole new turn. He'd grow up with a different set of values instilled in him. He might be more diplomatic from the start, having been exposed to Hiruzen's leadership style. His motivation to become Hokage might be less about seeking recognition and more about following in his father's footsteps. Also, his relationship with the village elders would be different. Instead of being an outcast, he'd be seen as the heir, which would bring its own set of challenges and opportunities for growth.
Perhaps he uses some ninja techniques like a shadow clone jutsu. He could make multiple clones to help him sort through and complete the paperwork quickly.