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.
One great Issei x Ophis fanfiction is 'A Bond Beyond Dimensions'. It beautifully explores their relationship in a way that stays true to the characters' personalities from the High School DxD series. The author does a wonderful job of developing their bond in different scenarios.
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 plot could be that Naruto and Hinata are transferred to the High School DxD world. Naruto's powerful ninja skills might make him stand out, while Hinata's shy yet strong - willed nature could attract different reactions. Maybe they have to face the devils and angels in this new world together, with Naruto using his Rasengan and Hinata her Gentle Fist techniques to protect their new friends and find a way back to their own world.
One common theme is power dynamics. Since 'godlike sans' is, well, godlike, it often changes the power balance in the 'High School DxD' world. Another theme could be the clash of cultures, as 'sans' comes from a very different universe compared to the one in 'High School DxD'.