His personality would be a mix of pride and a sense of superiority. He'd walk around the school as if he owns the place. However, he might also have a sense of honor. For example, if he made a promise to a character in the DxD world, he would keep it, but in his own haughty way. He might also be somewhat dismissive of the school's rules, believing that he is above them.
One possible plot is that Gilgamesh views the High School DxD setting as beneath him at first. But as he gets to know the characters like Issei and Rias, he starts to see the value in their struggles. He could then use his vast resources and power to help them in their fight against stronger enemies, all while maintaining his haughty and kingly demeanor.
In a Naruto and High School DxD fanfiction, there could be a scenario where the protagonists of both stories meet in a neutral dimension. Issei from High School DxD might be intrigued by Naruto's ninja skills and determination. Naruto, in turn, could be curious about Issei's Sacred Gear. They could start training together, sharing their knowledge and skills, and ultimately have to face a common enemy that threatens both their worlds.
He would likely be portrayed as arrogant. Given his status as a powerful and ancient figure in his original lore, in DxD fanfiction he would probably look down on most of the characters. He would act as if he is above the petty squabbles of the DxD world.
One possible fanfiction could be about Naruto characters attending the school in High School DxD. For example, Naruto's positive and determined attitude might clash with the more complex social and power dynamics in the DxD world. He could make new friends or even rivals among the devils, angels, and fallen angels.
One idea could be that Ben Tennyson somehow gets transferred to the world of High School DxD. He has to keep his alien forms a secret while dealing with the devils, angels, and fallen angels in the school. Maybe his Omnitrix could be misinterpreted as some kind of powerful devil device.
One possible fanfiction could be about Naruto characters somehow ending up in the world of High School DxD. Maybe Naruto's skills in combat and his determination would make him a formidable opponent in the battles that often occur in the High School DxD universe. He could form interesting alliances or rivalries with the characters there.