The relationship can be developed through shared experiences. For instance, they could go on a mission together where Naruto is still young and inexperienced. Kakashi would protect Naruto and Naruto would learn from Kakashi's skills and strategies.
In some fanfictions, Kakashi might be the one to help Naruto deal with the loneliness he feels because of being an orphan. He could take Naruto on little outings around the village, sharing stories about his own past and making Naruto feel like he has a real family with Kakashi as his daddy.
In Naruto fanfiction, characters like Jiraiya could also be a 'baby daddy'. He was known for his womanizing ways in the series, so it's not hard to fathom a situation where he has a child. Then there's Itachi. Although his life was full of tragedy and secrecy, a fanfiction could explore a scenario where he has a child before his untimely death. And Kiba could be a character as well. His close bond with his dog Akamaru and his brash personality could lead to a situation where he becomes a 'baby daddy' in a creative fanfiction.
One popular story could be 'Kakashi's Unexpected Fatherhood' where Kakashi discovers he's a father in a very unexpected situation. Maybe a former lover left a baby on his doorstep.
Fans often portray Kakashi as initially shocked but then very protective. He's usually shown as using his ninja skills to keep the baby safe, like creating special seals around the nursery.
Kakashi would approach Naruto's pranks with a mix of discipline and love. For instance, if Naruto tries to scribble on the walls, Kakashi would stop him gently. He'd then sit Naruto down and show him where he can draw instead, like on a piece of paper. He'd also tell Naruto stories about how important it is to respect their home. When Naruto does something really naughty, like trying to pull the cat's tail, Kakashi would firmly say 'No' but then immediately follow it up with a hug. He wants Naruto to learn right from wrong but also wants to make sure Naruto knows he is loved no matter what.
Iruka sensei is another possibility. He is kind and nurturing, having already shown his care for Naruto when he was a student. As a baby daddy, he would be patient and dedicated to raising his child with good values and a love for learning.
Kakashi would become more nurturing. In the original, he was a teacher, but as an adoptive father, he'd have to be more affectionate. He might show his face more often to Naruto, as a sign of trust, and be more open about his emotions.
Obviously, Danny Phantom himself could be the 'baby daddy'. His dual nature as a human and a ghost could bring a lot of interesting twists to the story of having a baby. For instance, the baby might inherit some of his ghost powers, which would create all kinds of challenges for raising the child.
The story could also focus on how Kakashi's decision to adopt Naruto affects the village's perception of Naruto. People might start to see Naruto in a different light, not just as the Jinchuriki but as Kakashi's son. This could lead to Naruto having a more positive experience growing up in the village, with less ostracism. Kakashi would have to navigate the politics of the village while also raising Naruto, which could add a lot of interesting twists to the plot.