Perhaps he could obtain the Rinnegan through some sort of ancient ritual or a gift from a powerful entity. And to become Hokage, he would need to prove himself worthy through his deeds and leadership skills, like protecting the village from a major threat.
In such a fanfiction, Naruto would be extremely powerful with the Rinnegan. He could use powerful jutsu like the Six Paths techniques. As Hokage, he would bring a new era of peace and strength to the village. His Rinnegan abilities might help in better defense against threats and also in diplomatic relations as other villages would fear his power.
Perhaps a plot where Naruto's Rinnegan is a result of a curse placed on him by an ancient evil. As Hokage, he has to not only fight against external threats but also the inner struggle of controlling the power of the Rinnegan without succumbing to the curse. He could learn about the origin of the Rinnegan from old scrolls and seek the help of sage-like characters to overcome the curse.
A new character could be introduced as a rival. Maybe a rogue ninja from a far - off land who also has a powerful dojutsu and sees Naruto's Rinnegan and position as Hokage as a threat to his own plans for power. This character could constantly try to undermine Naruto's rule and steal his Rinnegan powers.
Well, it depends on the specific fanfiction. Some fanfictions might explore alternate storylines where Naruto doesn't become the Hokage at all. Maybe he decides to pursue a different path in life, like focusing on being a teacher or an explorer. However, in a large number of fanfictions, he does achieve his goal of becoming the Hokage because it's such an iconic and desired outcome for fans who want to see his journey to the top in different ways.
It could be the First Hokage, Hashirama. Naruto would be amazed by his powerful Wood - Style jutsu and his ability to create the village. He could learn about the original ideals behind the Hidden Leaf from Hashirama.
One implication is that he becomes much more powerful. The Rinnegan gives him access to powerful abilities like the Six Paths techniques, which would change the power dynamics in battles in the fanfiction. He could now face stronger enemies more easily.
His friends like Sakura and Shikamaru would surely support him. Sakura, with her medical skills, could be an important part of his regime, providing healthcare for the ninjas. Shikamaru, being intelligent, could offer strategic advice. Also, the younger generation of ninjas who have been inspired by Naruto would be his strong supporters, as they believe in his leadership and the new direction he could take the village in.
Sasuke could be an alternative. He has strong ninja skills and a deep understanding of the ninja world. His experiences have made him grow into a very capable person, and his Sharingan gives him an edge in combat and strategic planning.
In Naruto fanfiction, the Rinnegan in Naruto is often portrayed in really cool and powerful ways. Some fanfics might give Naruto the ability to use all the Rinnegan powers to their fullest extent right away, like being able to effortlessly summon the Gedo Statue or use the Deva Path powers without any drawbacks. It's a way for writers to explore new storylines and make Naruto even more of a badass than he already was in the original series.