Well, in fanfiction, anything can be possible through creative writing. Maybe it could be possible through some sort of multiverse - hopping or a powerful magic that fuses the Naruto universe with the Saiyan world. It could also be that some unknown entity alters Naruto's genetics to make him a Saiyan.
One way for Naruto to be a full - blooded Saiyan in fanfiction could be a result of a divine intervention. Let's say a god or goddess from a higher plane of existence decides to experiment. They see Naruto's potential as a ninja and think, 'What if we added Saiyan blood to this?' So they manipulate the fabric of reality to give Naruto Saiyan heritage. As a Saiyan, Naruto would have a natural inclination towards combat and a drive to constantly get stronger. His training would take on a whole new dimension as he tries to master Saiyan abilities like ki control in addition to his ninja jutsu. His relationships might also change as his Saiyan nature makes him more aggressive at times, but his friends would help him find a balance.
If Naruto were a full - blooded Saiyan in fanfiction, his entire story would change. His growth would be different. Instead of chakra, he'd have ki. He could sense other powerful beings more easily. His training would likely involve more physical combat and less of the ninja - style training. Also, his Saiyan heritage might make him more of an outcast in the ninja world initially, as he'd be so different from everyone else.
Yes, he could. He could combine his Saiyan powers with ninja skills. For example, he could use his super speed to perform faster ninjutsu. His Saiyan strength could also be used in close - combat situations during ninja battles.
He would be more aggressive. Saiyans are known for their warrior - like nature, so Naruto might be quicker to anger and more eager to fight.
It could also be due to a strange prophecy. In both the Naruto and Dragon Ball worlds, prophecies play a big role. There could be a prophecy that a hero from one world would gain the powers of the other to defeat a common enemy. Naruto, being the hero of his world, is chosen and through some spiritual or magical transformation, he becomes a Super Saiyan.
One way to develop the story is to have Naruto face a new, extremely powerful enemy that his normal Naruto powers can't defeat. This forces him to seek out a new source of power, which leads to him becoming a Saiyan. Once he becomes a Saiyan, he has to deal with the cultural shock of the Saiyan way of life. He might struggle with the Saiyan's more aggressive and battle - hungry nature at first. But as he adapts, he starts to use his Saiyan powers to protect both his own world and the Dragon Ball universe. He could also form new friendships with the Dragon Ball characters, like Bulma who could help him understand his new powers better.
Well, in this kind of fanfiction, it might happen through some sort of cross - dimensional magic. Maybe there's a rift between the Naruto universe and the Dragon Ball universe that allows Naruto to gain the power of a Super Saiyan God. It could also be the result of a powerful wish made by a god - like character who decides to give Naruto this new form for the sake of saving his own universe or something like that.
In the fanfiction, perhaps there is a prophecy that Naruto will become a Super Saiyan God - like being. As he gradually transforms, his body and mind change. He starts to see the world in a different way. His new powers attract the attention of both friends and foes. His friends try to help him adjust, while his foes try to either steal his power or destroy him. The plot could revolve around Naruto's journey to master his new form, protect his loved ones, and save the ninja world from destruction that only he, with his new powers, can prevent.
If Naruto was born a Saiyan in fanfiction, he'd likely have super strength. He could easily lift boulders or break through solid walls with his bare hands. His speed would also be incredible, able to move faster than the eye can see. And of course, he'd have the ability to fly, which is a common Saiyan ability. His ki blasts would be a powerful weapon, able to blast away opponents from a distance. He might also have the Zenkai boost, getting stronger every time he recovers from a near - death experience.
One possible plot could be that Naruto discovers he has Saiyan blood somehow. He then starts to develop Saiyan powers like super strength and speed. He has to hide these new powers from his friends at first because he doesn't understand them. As he learns more about his Saiyan heritage, he becomes stronger and eventually reveals his new form to protect the village from a powerful new enemy.
Well, in a Saiyan Naruto x Bulma fanfiction, the story might start with Naruto being sent to Bulma's world due to some sort of interdimensional rift. Since Naruto is a Saiyan in this concept, he might have an immediate connection with Bulma because of his unique power. Bulma, being the curious scientist she is, would be intrigued by Naruto's Saiyan abilities and they could go on adventures together, exploring new forms of energy or technology. They could also form a bond over protecting the world from threats that neither of their original worlds had faced before.