At first, Jiraiya would be in shock. He couldn't believe that Naruto has become so dark and left him. Then, he would be determined to bring Naruto back. He would follow Naruto's trail, trying to understand his motives. Along the way, he might face some challenges from Dark Naruto's new allies or the places he visits. But Jiraiya would not give up easily, as he truly cares about Naruto and wants to save him from the darkness.
It could be Orochimaru. Since Orochimaru has a long - standing feud with Jiraiya and is always up to no good. If Naruto ditches Jiraiya, Orochimaru might see this as an opportunity to manipulate Naruto. He could try to convince Naruto that Jiraiya was holding him back and that he can offer Naruto more power, all while having his own evil plans for Naruto.
Naruto would first be extremely confused by the rules of the Death Note. His understanding of power and its consequences is rooted in his ninja training and his own moral compass. The Death Note's rules, which seem so simple yet so deadly, would be completely foreign to him. He would question why anyone would create such a thing and would be very hesitant to even consider using it, as it goes against everything he stands for in his world of ninjas, friendship, and protecting the innocent.
In fanfiction, Jiraiya seeing Naruto as Hokage can be a very emotional moment. It could be a vision Jiraiya has in a story where he's still alive, or perhaps it's a memory in a story set after his death. This often shows Jiraiya's pride in Naruto and his belief in Naruto's potential to lead the village.
They often make him seem irresponsible. For example, in normal Naruto story, Jiraiya is seen as a mentor to Naruto. But in bashing fanfiction, he may be shown as being more interested in his own adventures and leaving Naruto alone at crucial times.
In the Naruto universe, bringing someone back from the dead is extremely difficult. While there are some jutsu like the Edo Tensei that can reanimate the dead, it has its limitations and is often considered dark magic. For Naruto to revive Jiraiya, he would need to find a way around the natural laws of life and death. However, in fanfiction, anything is possible as it allows for creative interpretations and bending of the established rules.
He might be more confident from the start. Since Jiraiya is known for his charisma and self - assurance, Naruto could inherit some of that. He may not be the underdog we saw in the original series as much.