In a fanfiction where Naruto and Hinata join Orochimaru, one possibility is that they are forced into it. Maybe Orochimaru has some kind of leverage over Konoha that makes them comply. Naruto, being the head - strong character he is, would likely be constantly looking for a way to escape and take Hinata with him. Hinata, on the other hand, would probably be more reserved and observant, trying to find Orochimaru's weaknesses.
On the other hand, their relationship could also grow stronger. They would be relying on each other more in this new and dangerous situation. Naruto would be even more protective of Hinata, and Hinata would be a source of comfort for Naruto. They would have to communicate and trust each other more deeply to survive in Orochimaru's lair.
Well, if Naruto saves Hinata from Orochimaru in a fanfic, it might start with Hinata being captured by Orochimaru's minions. Naruto would immediately go on a rescue mission. He'd probably fight through Orochimaru's various snake - based defenses. And when he finally reaches Hinata, he'd defeat Orochimaru with a powerful Rasengan, and then carry Hinata to safety.
In a fanfiction where Naruto and Hinata join Kumo, one possibility is that they are sent there on a diplomatic mission that turns into a permanent stay. Naruto's charisma could win over the Kumo villagers, and Hinata's Byakugan would be a valuable asset. They could work together to reform some of Kumo's more archaic shinobi practices. Naruto could also train with the Kumo ninjas, sharing his sage mode knowledge, while Hinata could teach them about the Hyuga clan's gentle fist techniques.
One possibility is that Naruto discovers Orochimaru's lair through some clever detective work. He then sneaks in, avoiding all the traps. When he sees Orochimaru about to perform some sort of experiment on Hinata, Naruto jumps in and starts fighting. He might transform into his Nine - Tails Chakra Mode to gain an edge in the battle and save Hinata.
Perhaps Naruto teams up with some unexpected allies. For example, he could convince some of Orochimaru's former henchmen to turn against him. These henchmen could provide valuable information about Orochimaru's lair and the security measures in place. Using this information, Naruto can plan a strategic rescue and save Hinata.
There are fanfictions involving Orochimaru and Hinata. Some might explore unique storylines where Orochimaru has an unexpected interaction with Hinata. Maybe he could be interested in her Byakugan for some strange reason in the fan - made stories. It really depends on the creativity of the authors. You can search for tags related to them on fanfic platforms to discover relevant works.
One possible motive could be the pursuit of power. Orochimaru is known for his vast knowledge of jutsu. Naruto might think that by joining him, he can gain the strength needed to protect his village better or to bring Sasuke back more effectively.
Well, in such a fanfic, Naruto joining Orochimaru would be a huge shock to the Naruto -verse. Orochimaru has always been an enemy, so for Naruto to side with him would mean a major shift in the power dynamics. Naruto could learn a great deal about the darker side of the ninja world, but he'd also be in constant danger of being used by Orochimaru for his own purposes. It would be a very different Naruto story, full of moral dilemmas and unexpected alliances.
In this fanfiction setting, their relationship would likely evolve in unique ways. Being in Kumo, they'd be in a different social environment. They might have to face some prejudice as outsiders, but this would make them cling to each other more. Naruto would be more protective of Hinata, and Hinata would be Naruto's source of strength. They'd also have more private time together, which could lead to a more romantic and intimate relationship.