Obviously, Naruto himself would be a key character as the story focuses on his serious training. Kakashi could also be important. He might be Naruto's guide in this new training regime, sharing his own experiences and wisdom to help Naruto grow stronger.
In the 'naruto trained seriously fanfiction', Naruto could be training with his friends in a more serious way. They might form a special training group where they push each other to their limits. For example, they could be training to improve their teamwork and combination jutsus. Naruto might focus on his speed and strength training while also learning how to better coordinate with his friends' abilities, like Sasuke's Sharingan - based attacks and Sakura's medical and combat skills.
One possible plot could be Naruto making great leaps in his ninja skills. He might learn powerful jutsu more quickly under Jiraiya's intense training regime. For example, he could master new forms of Rasengan faster and also gain better control over his chakra.
It could be a rogue ninja from a forgotten clan. This ninja might have a vendetta against Naruto's village and has powers that are difficult to counter. So, Naruto has to train to kill him to protect everyone.
Naruto would likely become more disciplined. Jiraiya's strict training would force Naruto to focus and follow a routine, which is different from his usual impulsive self.
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.
A likely main summon could be Gamabunta. He's big, powerful, and has a long - standing relationship with Naruto. He could teach Naruto some really advanced water - based jutsu.
It could be a powerful dark wizard who wants to control the dragons and sees Naruto as a threat because of his newfound dragon - trained powers.
Obviously, Naruto and Asuma would be the main characters. Naruto, the protagonist of the Naruto series, is the one being trained. And Asuma, a respected jonin in the Hidden Leaf Village, is the trainer.
The main character is Naruto without a doubt. He is the one on the path to becoming god - like through the Kyuubi's training. The Kyuubi is another key character. It has a vast amount of knowledge and power, and it imparts this to Naruto. Additionally, characters like Jiraiya could be important. Since Jiraiya was Naruto's mentor in the original story, he might play a role in this fanfiction too, perhaps guiding Naruto in a different way in combination with the Kyuubi's training. And then there are Naruto's friends. Sakura could be there to support Naruto or be in awe of his new powers. Sasuke might be an interesting character as he could be Naruto's rival, either trying to catch up to his power or oppose him if their paths diverge due to Naruto's god - like transformation.
In addition to Naruto, there could be other characters from the Naruto universe who might get involved in his training with the Dragon Clan. For instance, his friends like Sasuke or Sakura might come to check on him or even be part of some joint training sessions. And there could be rival characters within the Dragon Clan who don't like Naruto being there and cause trouble for him during his training.