Well, Kakashi might start by spending more time with Naruto. He could train Naruto more intensively, like teaching him new jutsu. Maybe he also shares his own experiences to give Naruto some guidance and motivation.
In such a fanfiction, Naruto might become extremely powerful under Kakashi's training. He could learn new jutsus faster and gain better control over his chakra. Maybe he'd also develop a closer bond with Kakashi, different from his relationship with Iruka or other characters in the original story.
Maybe Kakashi is preoccupied with other important missions. He is a highly - skilled and often - needed ninja. There could be some top - secret missions that require his full attention, leaving him no time to train Naruto properly.
Naruto might also try to find alternative ways to gain strength. He could explore the forest around the village to find new training areas or discover new chakra - related phenomena. He has a curious nature, and this could lead him to stumble upon something that gives him an edge in his training, like a hidden hot spring that has special chakra - enhancing properties.
A plot development could be that Kakashi discovers the root cause of Naruto's neglect. It might be related to some hidden secrets in the village. Then Kakashi has to fight against those who are behind Naruto's neglect while still training Naruto. Another development could be that Naruto and Kakashi form a very strong bond. Naruto then uses the skills and confidence he gained from Kakashi to face his enemies and also to try and change the village's attitude towards him.
Often, there's a plot point where Naruto discovers a hidden talent or power on his own because Kakashi isn't there to guide him. He might also get into more trouble than usual as he tries to prove himself without Kakashi's approval. And sometimes, the other members of Team 7, like Sasuke and Sakura, start to notice Kakashi's neglect of Naruto and try to make up for it in their own ways, which can create some interesting group dynamics in the fanfiction.
Well, in such a fanfiction, Naruto might become more self - reliant. Since he's being neglected by Kakashi, he has to find ways to train and grow on his own. This could lead to him developing a tougher exterior and being more determined to prove himself.
In this fanfiction, Naruto's relationship with his classmates would shift. Since he's being trained intensively by Kakashi, some might be jealous of the special attention he's getting. But others might respect him more for his growth. His bond with Shikamaru could also change. Shikamaru, being perceptive, would notice Naruto's newfound confidence and skills. They might end up collaborating more in missions or battles, with Naruto taking on a more leadership role in some cases.
One common plot is Naruto seeking alternative mentors. Since Kakashi is neglecting him, he might look for other ninja in the village who can teach him. Another plot could be Naruto trying to prove himself to Kakashi by achieving great feats on his own. Some fanfics also show Naruto developing a closer bond with other outcasts in the village as he feels neglected by Kakashi, who is supposed to be his teacher.
One common plotline is Naruto seeking out other mentors. Since Kakashi is neglecting him, he might look for someone like Jiraiya earlier than in the original story.