His relationship with Sasuke would become extremely complicated. Instead of being rivals - friends, Sasuke would see Naruto as a threat. He might try to fight Naruto to bring back the old Naruto or to stop his tyranny. Sakura would be horrified at Naruto's transformation. She might initially try to reason with him, but if that fails, she could become one of his opponents. His relationship with his former sensei Kakashi would also be strained as Kakashi would not recognize the Naruto who has become Darth Vader.
His relationships would change drastically. With Hinata, who had feelings for him, she would be in despair. She might try to reach out to the good in Naruto, but it would be difficult. His friends from Team 7 would be divided. Some might believe there is still hope for Naruto and try to redeem him, while others would see him as a lost cause. And his relationship with the village elders would also change. Instead of being a hero in their eyes, he would be a dangerous figure they need to either control or eliminate.
His ability to move would change drastically. Vader has his slow, yet intimidating, mechanical walk. Naruto might adopt a similar gait as he becomes more like Vader. In terms of combat, Naruto's Rasengan could be combined with the dark side of the Force. It could become a more destructive and corrupting attack, maybe with a black, swirling energy around it. Also, his ability to sense others, which was based on chakra before, could now be a more Force - like sense, allowing him to detect enemies even through walls and over long distances.
Well, the plot could be that Naruto, through some dark magic or strange time - space anomaly, gets the powers and persona of Darth Vader. He might start ruling the Hidden Leaf Village with an iron fist, suppressing any form of opposition. His friends would be shocked and some might try to bring the old Naruto back. It would be a total shift from his usual positive and friendly self to a dark and menacing figure.
His relationship with Sasuke would be really interesting. Since Sasuke has his own complex connection to the Uchiha clan and Madara, Naruto becoming a Madara fan could either bring them closer together in understanding the Uchiha's past or drive them further apart if Sasuke doesn't agree with Naruto's new - found admiration. Maybe Naruto would start to push Sasuke towards a more 'Madara - like' view of power and revenge.
His relationship with Sakura would change a lot. Sakura might be initially scared of Naruto's fox form, but as she realizes it's still him inside, she would start to see him in a new light. She could become more protective of him. With Kakashi, Kakashi would be more cautious and observant, trying to figure out how to help Naruto control his new form. And with his rivals, they might either be more envious or more afraid of him now that he has this unique transformation.
His relationship with Sasuke would be very different. Sasuke might see Naruto as even more of an enigma. He could be both jealous of Naruto's new power and also concerned about what Naruto has become. Naruto, on the other hand, would try to show Sasuke that he's still the same person deep down.
Naruto's relationship with the villagers could be affected. As he becomes a Minato fan, he would try to uphold Minato's legacy even more. The villagers, seeing Naruto's newfound admiration for Minato, might start to see Naruto in a new light, more like how they saw Minato. Also, his relationship with Hinata could be different. Naruto might start to model his family - centered values after Minato, which could bring him and Hinata even closer as they plan their own family.
His relationships would change drastically. With his friends in the Hidden Leaf, there would be a sense of distance. For example, Sasuke might be confused by Naruto's decision. But at the same time, Naruto might form new relationships with the animals and plants in his hermitage area.
His relationship with Sasuke would become very strained. Sasuke, who has always been somewhat distrustful, would see Naruto as a real threat now. He might even consider Naruto an enemy and try to stop him.
His relationship with Sakura would change a lot. Sakura might be more protective of Naruto as a cat, seeing him as a cute and vulnerable creature. She might take on a more motherly role towards him. With Kakashi, Kakashi would probably be amused by Naruto's cat form at first, but then he'd start to worry about how to turn him back. And as for his friends in general, they'd have to communicate with him in different ways, like through body language and meows.
His relationship with Sasuke would be greatly affected. Sasuke might see Naruto as a completely different person now that he has become the Nine - Tails. There could be a sense of fear or at least a new level of caution from Sasuke. They might have to re - establish their bond, as Naruto's new form could be a threat in Sasuke's eyes.