There could be characters like Shikamaru. He might be interested in the new techniques Naruto is learning under Kakashi's training and try to analyze them. Kiba and Akamaru could also be side characters. They might challenge Naruto to test his new - found skills. And of course, the Hokage at that time, whether it's Tsunade or someone else, could keep an eye on Naruto's development as it could be important for the village.
Sasuke could be a side character. Since Naruto is getting special training from Kakashi, Sasuke might feel jealous or motivated to train harder himself. Sakura could also be there, perhaps she tries to learn from Kakashi's training methods to help Naruto better. Another potential side character could be Iruka, who might check on Naruto's progress from time to time and offer some words of encouragement.
Well, it could be that in this fanfic, Kakashi discovers a new potential in Naruto. He might train Naruto in some ancient and forgotten ninja arts. This would set Naruto apart from his canon self. Also, their relationship could be explored more deeply, like Kakashi becoming a father - figure to Naruto, and this emotional bond could influence Naruto's growth as a ninja.
The Otsutsuki clan members other than Kaguya could be antagonists. Since Kaguya is training Naruto, they might see Naruto as a threat to their plans.
Orochimaru could still be an antagonist. He might see Naruto as a threat especially if Kakashi is training him intensively. His goal of obtaining power could lead him to target Naruto.
He might gain better chakra control. Kakashi could teach Naruto how to precisely manage his chakra, allowing him to perform jutsus more efficiently and for longer periods. For example, Naruto could use his Shadow Clone Jutsu without exhausting his chakra so quickly.
Well, Kakashi might introduce Naruto to more strategic ways of fighting. For example, he could teach Naruto how to better analyze his opponents before attacking. Naruto, who was previously more impulsive, may start to hold back and look for the right moment to strike, like Kakashi does with his Sharingan.
Kakashi could be another one. Since he's already Naruto's teacher in the original story, it's not a stretch to think he could start training Naruto even earlier in the fanfic. He could focus on different aspects of Naruto's abilities, like improving his stealth or his chakra control even more.
Well, Iruka - sensei could be a supporting character. He might check in on Naruto from time to time to see how he's doing under Kakashi. Tsunade could also play a role. Maybe she has some special tasks for Kakashi and Naruto as part of Naruto's apprenticeship. And then there are Naruto's other friends like Choji and Kiba. They could be involved in group training sessions or missions that are related to Naruto's training with Kakashi. For example, they could be the ones Naruto has to protect during a mission as a test of his skills learned from Kakashi.
Well, in such a fanfiction, Naruto might be trained by someone else entirely. Maybe he gets trained by Jiraiya right from the start more intensively. This could lead to Naruto having a different fighting style, perhaps more focused on toad - related jutsus earlier on.
Iruka - sensei. He was Naruto's first teacher and has a connection to Kakashi as well. He could provide some emotional support or some important backstory information that helps Naruto and Kakashi in their journey in the fanfic.
Another option could be Itachi. Although Itachi is from a different village and has his own complex situation, in a fanfiction scenario, he could secretly train Naruto. Itachi has extremely powerful jutsus and a deep understanding of the Sharingan. Training under him would give Naruto a very different perspective on power and combat, not to mention some insights into the Uchiha clan.