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.
Naruto's character would become more mellow. Kurenai's gentle and nurturing nature would have a calming effect on him. He would be less impulsive. And with Asuma as a role model, Naruto would become more responsible. He would learn the importance of family and loyalty from them.
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 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.
Well, it could be a story full of mentorship. Asuma could teach Naruto his unique jutsu, like the Wind Style. Naruto would also learn about Asuma's values, such as loyalty and perseverance. And perhaps they would bond over a shared love for food. Maybe Naruto would become closer to the Sarutobi family in general.
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.
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.
He would likely become more secretive. Since ANBU operates in the shadows, Naruto would have to keep a lot of things to himself, which is a big change from his usually open - minded self. Also, the death of Asuma could make him more ruthless in combat, as he realizes the harsh reality of the ninja world. He may still hold onto his core values of protecting his friends and the village, but his methods could be more extreme in the ANBU.
His relationship with Sasuke would be different. Sasuke might view Naruto's connection with the Fem Kyuubi as a new source of power and be more wary or even jealous. But Naruto could use his new situation to try and bring Sasuke back to the right path earlier.