His relationships with his old friends would be different. Some of his friends might be distrustful at first. For example, Sasuke might see Naruto's new situation as a betrayal of their shared experiences. However, over time, if Naruto could prove that he was still the same person at heart, they might reconcile. With Orochimaru as his adopted father, Naruto would also form new relationships within Orochimaru's circle. He could meet some of Orochimaru's former subordinates who might now be on a more positive path, and they could become comrades or even friends.
Naruto's relationship with the village would be strained. The village elders and many of the villagers would be cautious about Naruto being associated with Orochimaru. But Naruto could use this opportunity to bridge the gap between the village and Orochimaru's faction. He could work towards a more united ninja world. In terms of his relationship with his teachers, they might be concerned about Orochimaru's influence on Naruto, but if Naruto showed growth and positive development, they could come to accept the situation.
Naruto's character would change a great deal. He would be exposed to Orochimaru's world of experimentation and dark magic. This might make him more cynical about the ninja world. Also, he would have to adapt to Orochimaru's strict and often cruel training methods. As a result, he could become more introverted and less trusting of others. His once open - hearted nature might be replaced with a more guarded one.
If Naruto was raised by Orochimaru in fanfiction, his relationships would be turned upside down. His friendships from the Leaf Village would be in jeopardy. Since Orochimaru is not one to value the kind of loyalty and friendship that Naruto's friends in the Leaf do, Naruto would have a hard time understanding and maintaining those relationships. He might even be hostile towards them, seeing them as obstacles in his pursuit of power. His relationship with the Hokage would also be a mess. Instead of looking up to the Hokage as a role model, he would probably view the position as something to be conquered or undermined, following Orochimaru's ideology of power - seeking.
His relationships with his friends in Konoha would become more complicated. They might be suspicious of Medusa and worry about Naruto being influenced by her. But at the same time, Naruto could try to convince them that Medusa is not all bad. With his enemies, they might be more afraid of him because of his connection to Medusa.
His relationship with Hinata would be completely different. Instead of a potential love interest, she'd be his sister. His relationship with Neji might also change. Neji might be less hostile towards Naruto as they'd be part of the same family. Naruto would also have a new relationship with Hiashi as his adoptive father, and he'd have to follow Hiashi's rules and guidance.
His relationship with his old friends in the village would become more complex. They might see him as someone who has moved into a different social class. For example, Sasuke might view him with suspicion or envy, thinking Naruto has abandoned their shared past as shinobi.
His relationships would change a great deal. He'd probably form a father - son type of bond with the Raikage. With the other Cloud ninjas, he'd start as an outsider but gradually earn their respect through his skills and determination. His relationship with his friends in Konoha might become more complex as he now has a new allegiance to the Hidden Cloud.
Naruto's relationship with the Hokage would also be altered. He might not have the same drive to be recognized by the Hokage in the same way. Instead, he would be more concerned with making Inoichi proud and living up to the Yamanaka clan's expectations. And his friendships within the Yamanaka clan, like with Ino, could be really deep. They might train together and develop new jutsu as a team, which is different from his relationships in the original story.
His abilities would likely take a very different turn. Orochimaru could introduce him to some dark and powerful chakra - based techniques. Naruto might learn to manipulate his chakra in more insidious ways, perhaps using it to control others or for stealth attacks.
His relationship with Sasuke would be different. Since Naruto would have a more stable upbringing, he might approach Sasuke with more understanding and less of the desperate need to bring him back. His friends in Konoha would also see him in a new light, as he is now under the care of two very respected shinobi. And his relationship with the village elders might be less strained, as Tsunade could act as a buffer.
His relationship with Sasuke might improve. Since Kakashi would be his adoptive father, he could better mediate between Naruto and Sasuke. Also, with Kurenai around, Naruto might become closer to Hinata as Kurenai could encourage their friendship. And his relationship with the other Konoha 11 would be different too. He'd be part of a family that has strong ties in the village, so they might look at him in a new light.