He could become more of a lone wolf. Instead of relying on his friends as much, he might feel that he has to take on the burden of saving the world all by himself. His once - open - hearted nature could be replaced with a more guarded one. He'd be constantly on the lookout for threats, both from the apocalyptic forces and from other survivors who might want to take advantage of him.
Naruto's normally positive and upbeat character might become more hardened. He'd be more cynical after seeing so much destruction. His trust in others might be shaken, especially if he's been betrayed during the apocalypse.
He would become more cynical. Instead of his usual optimistic self, he'd view the world in a darker light. His trust in people would be severely damaged.
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 personality might become more cautious. Since he has a greater responsibility as a guardian, he can't be as reckless as he was sometimes in the original Naruto series.
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.
Well, Naruto's powers would likely change quite a bit in a Naruto becomes Menma fanfiction. Menma is shown to have a more natural and effortless control over his abilities. So Naruto might find that his Rasengan becomes easier to form and manipulate. Also, his ability to sense danger and opponents might be enhanced in a different way than before. His connection to the natural energy around him could be more in tune with Menma's style, leading to new and different applications of his existing powers.
His personality might become more withdrawn. As a lich, Naruto could be feared by others, and this might make him more hesitant to interact with people. He was always so outgoing before, but now he might keep to himself more.
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.