Naruto's development would be a journey of self - discovery. As Orochimaru's apprentice, he would be exposed to different forms of power and knowledge. He might initially be disgusted by Orochimaru's methods but then realize that he can use some of it for good. His social skills would change too. He would be more adept at dealing with enemies and those with different motives. His sense of humor might even change, becoming darker to fit in with Orochimaru's crowd. But deep down, his core values of friendship and protecting the weak would still be there, just masked by his new persona as Orochimaru's apprentice.
The storyline could be very dark. Naruto, as Orochimaru's apprentice, might be forced to learn some forbidden jutsu. He would probably be in a constant moral struggle. Orochimaru is known for his unethical experiments, and Naruto could be made to assist in them. However, Naruto's inherent goodness might lead him to try and subvert Orochimaru's plans from the inside. Maybe he would pretend to go along with Orochimaru's ideas while secretly finding ways to undermine his power and free the other captives in Orochimaru's lair.
Naruto would likely become more secretive. Since Orochimaru is a very private and often devious character, Naruto might pick up some of those traits. He would probably keep things from his friends more often.
They would develop in terms of their skills. Under Orochimaru, they would gain a lot of knowledge about chakra manipulation and forbidden jutsu. But with Naruto, they would learn about the power of bonds and friendship. This combination could lead to the development of new jutsu that combines both physical power and the strength of relationships. Also, they would grow in their understanding of right and wrong, as Naruto's moral compass would constantly be in conflict with Orochimaru's more amoral stance.
Naruto's character would likely be darker. Orochimaru is known for his unethical experiments and thirst for power. Naruto, under his tutelage, might be more focused on individual strength rather than the bonds he values in the original series. He could become more secretive and less trusting of others.
The apprentice could learn unique jutsu from Orochimaru. Since Orochimaru is known for his forbidden jutsu, the apprentice might gain an edge in combat but also face the danger of being corrupted by the power. They could then use these skills in battles against common enemies. And, in the process, they might try to redeem Orochimaru in some way, like showing him that there is a better path than his thirst for power.
He might learn Orochimaru's signature snake - related jutsu, like the Hidden Shadow Snake Hands. This jutsu allows the user to summon snakes from their sleeves or other hiding places to attack opponents.
In fanfic, anything can be possible, so Orochimaru could be Naruto's father. But in the official Naruto storyline, it's not the case. Naruto's parentage is clearly defined as Minato and Kushina. Fanfic writers might be trying to explore a different kind of family drama or a new take on the characters' relationships by creating this unorthodox connection.
One possible fanfic could be about a student of Orochimaru who rebels against his evil ways. This student, perhaps having seen the destruction Orochimaru causes, decides to use the knowledge he has gained in a positive way. He might start by secretly helping the Hidden Leaf Village, using his skills in jutsu and forbidden arts to protect rather than harm.
In some fanfics, Orochimaru can be made Naruto's sensei. Fanfic authors often like to play with the original story's elements and create unique scenarios. They might think that having Orochimaru as Naruto's sensei would lead to a very different kind of character development for Naruto, full of more moral and ethical challenges due to Orochimaru's dark nature.