Naruto might be initially very disappointed. He has always looked up to Kakashi as a strong and cool ninja, and being refused by him would be a big blow. He could become a bit sulky and withdrawn, feeling like he's not good enough.
One possible reason could be that Kakashi might be haunted by his past experiences. He has lost many comrades, and perhaps he's afraid that getting too close to Naruto and training him would put Naruto in danger as well. He might think that Naruto, being the Jinchuriki of the Nine - Tails, already has a target on his back, and training him would only make that target bigger.
Naruto might be initially very disappointed. He has always looked up to Jiraiya and expected to be trained by him. So, when faced with the refusal, he could be heartbroken and feel like he has lost a great opportunity.
Well, Kakashi could train Naruto in combat skills specific to ANBU missions. This would include more lethal and precise attacks. He might also teach Naruto how to use different types of weapons that are common in ANBU operations. Maybe Kakashi would set up mock battles where Naruto has to face multiple opponents at once, just like in real ANBU scenarios.
In some fanfictions, Kakashi might train Naruto by creating extremely challenging survival scenarios. For example, he could drop Naruto in a mock battlefield filled with various traps and enemies that Naruto has to overcome using his skills and wits. This forces Naruto to think on his feet and improve his combat abilities quickly.
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.
Kakashi would probably focus more on stealth training. In the original, Naruto was more about big, flashy attacks. But Kakashi would teach Naruto how to move quietly and strike unexpectedly. He might also teach Naruto more about chakra control from the very beginning. In the original, Naruto struggled with this for a while. And Kakashi would surely introduce Naruto to some of his own unique jutsu earlier, like the Chidori, but with Naruto's own spin on it.
Another reason could be that in this fanfiction, the plot is set in a way that there is a new training system in the village. And this system doesn't involve Kakashi in Naruto's training. There could be new mentors or a completely different approach to training young ninjas like Naruto, which excludes Kakashi from the equation.
One possible reason could be that Naruto has some kind of misunderstanding with Kakashi. Maybe he saw Kakashi do something that he misinterpreted, and it made him so angry or disappointed that he refuses to call him sensei.