It varies from one fanfiction to another. Some fanfictions might depict Naruto as dead to introduce a new set of challenges for the remaining characters. For example, if Naruto dies, Konoha might be left vulnerable, and characters like Sasuke or Sakura would have to step up in different ways. But there are also plenty of fanfictions where Naruto remains alive and continues his journey, perhaps facing new and even more dangerous threats than in the original series.
One consequence could be that it disturbs the balance of the ninja world. The dead are supposed to rest, and by controlling them, Naruto might be messing with the natural order. This could lead to other ninja clans or villages seeing him as a threat.
Kakashi could be another important character. As a mentor figure, he might be guiding the younger generation through the difficult times after Naruto's death, using his experience and wisdom to keep the village stable.
In some fanfics, it could be due to a forbidden jutsu he discovers. Maybe he stumbles upon an ancient scroll that details the process of controlling the dead.
In some fanfiction concepts, it could be due to a new, powerful jutsu he discovers or perhaps as a result of some ancient curse or blessing that is bestowed upon him.
It could be a rogue ninja from another village who has a vendetta against Naruto. This ninja might have set a very elaborate trap that made it seem like Naruto was killed and left his body in a place where his friends would find him and think he was dead.
Maybe it's his father, Minato. Minato has a lot of things to tell Naruto, like more about the Uzumaki clan and how to better use the power of the Nine - Tails. And as his father, their connection is very strong, so it's quite possible that he is among the first dead Naruto talks to.
Naruto's experience in dealing with tough opponents in his own world would help him in the zombie - filled Highschool of the Dead. He would be able to quickly analyze the situation, figure out the weaknesses of the zombies, and come up with effective plans. His friendship - first approach might also lead to some interesting relationships with the other characters in this new and dangerous world.
In this Naruto - Dead Rising fanfiction, Naruto could team up with some of the survivors in the Dead Rising world. They would be initially skeptical of his strange powers but soon realize his value. Naruto could teach them some of his ninja training techniques to help them better defend themselves against the zombies. And in turn, they could teach him about the unique aspects of the Dead Rising world, like how to find and use the best weapons against the undead.