The plot could involve Naruto being shunned by the Hidden Leaf Village for some reason and Kisame, seeing an opportunity, takes him in. Kisame might see Naruto as a tool at first, but as time passes, he develops a sort of father - son bond with him. Naruto would then have to deal with the fact that he's now associated with the Akatsuki, which would put him at odds with his former friends. He could use this new position to try and change the Akatsuki from within, with Kisame's begrudging support.
Naruto would likely become more reserved. Since Kisame is not as expressive as some of Naruto's friends in the Leaf Village, Naruto would pick up on that. His fighting style would also change. He'd probably be more stealthy and calculating, learning from Kisame's experience. For example, instead of his usual head - on attacks, he'd start to use more misdirection and ambush tactics.
There could be. Fanfiction is a vast realm where all sorts of ideas are explored. However, I'm not sure if there is a highly popular or mainstream 'naruto is kisame apprentice' fanfiction. It might be a more niche concept, but it's entirely possible that some creative writer has delved into this idea to create an interesting story about Naruto learning from Kisame, perhaps incorporating Kisame's unique fighting style and view of the ninja world into Naruto's growth.
Iruka could be an important character. He's always been like a father figure to Naruto, and now with this new revelation, he might have to help Naruto deal with it. Hinata could also be in the story, and her feelings for Naruto could be complicated by his connection to Kisame. And then there could be some original characters created just for this fanfiction, like a ninja who was a former ally of Kisame and has some information for Naruto.
Kisame would probably be very angry at first. Samehada is his weapon and has been with him for a long time. He might feel betrayed and try to get Samehada back by force.
One idea could be a story where Naruto and Kisame are forced to team up during a mission that has a greater threat to the ninja world. They start off distrusting each other but gradually come to respect one another's skills.
At first, it would be full of suspicion. Naruto is a Konoha ninja and Kisame is from the Akatsuki. But as the teaching progresses, they might develop a mutual respect. Naruto would respect Kisame's power and knowledge, and Kisame might respect Naruto's determination.
Well, in some fanfictions, Naruto might defeat Kisame in a battle and take the sword as a trophy. Maybe he uses his newfound Sage Mode powers or some other advanced jutsu to overpower Kisame and claim the sword.
In the fanfiction, it could be that Kisame gives Naruto Samehada as part of a deal. Maybe Naruto has to do something for Kisame in return. Naruto might struggle with the idea of making a deal with an Akatsuki member but decides it's worth it for the power of Samehada. Once he has it, he might face opposition from his friends and village elders who are suspicious of his new weapon and the way he obtained it.
In this fanfiction, perhaps Naruto is on a quest to learn more about Kisame. He could meet people who knew Kisame and through them, uncover secrets about his supposed father. This could lead to him having to make difficult decisions, such as whether to follow in Kisame's footsteps in some aspects or completely break away and forge his own path as a ninja.