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.
With Kakashi, it would be a complex situation. Kakashi might have tried to prevent Naruto's banishment but failed. When Naruto becomes Raikage, Kakashi would have to respect his new position. Naruto's relationship with the people of the Land of Lightning would start off as a bit of an outsider but as he becomes Raikage, they would look up to him. His relationship with the other Kage would also change. They would see him as a new player in the political and ninja world. Some might be skeptical of his sudden rise in the Land of Lightning, while others might be interested in forming alliances with him.
The relationship between Naruto and the Raikage would be completely transformed. Instead of being just an ally or a foreign ninja, Naruto would be his son. This would likely lead to a more protective and perhaps more demanding relationship from the Raikage. Naruto's relationship with Killer Bee might also change. They could become more like brothers, with Bee helping Naruto to understand his new identity and the culture of 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 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 the Leaf Village would become more personal. He'd be part of the Sarutobi clan in a sense. His bond with Shikamaru and Choji would strengthen as they are Asuma's students. Naruto would also have a new father - figure relationship, which would change how he views authority.
In this fanfiction, Naruto's abilities would change significantly. Firstly, he'd be exposed to Kumogakure's unique jutsu. He might learn more advanced lightning - style techniques which are prevalent in the Cloud Village. His combat style would be more brute - force - oriented, as the Raikage values strength. Also, his ability to lead and make strategic decisions would be molded by the Raikage's military - minded approach. He might not rely as much on his Uzumaki heritage in the beginning, but as he grows, he could find a way to combine the Raikage's teachings with his own innate abilities.
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 Sasuke might be different. With Kurenai's influence, Naruto could be more understanding and patient, which might help him better deal with Sasuke's issues. His bond with Sakura could also change as he may approach her in a more mature way, having learned from Kurenai about relationships.
Well, in this kind of fanfiction, Naruto might gain a whole new set of skills. The Raikage is known for his strength and tough training methods. So Naruto could become even more powerful in combat. He might also be exposed to a different culture within the Hidden Cloud Village. And perhaps he'd form new and interesting relationships with the Cloud ninjas.