One main element is his difficult upbringing in poverty - stricken surroundings. Another is the constant bullying and discrimination he endured due to his physical appearance. And of course, his strong bond with his sister Daki. This bond was what drove him to protect her at all costs and also influenced his actions as a demon. For example, he would go to great lengths to avenge any harm done to her because of their shared experiences of being outcasts in society.
Gyutaro was born into poverty. He and his sister Daki had a very tough life. They were constantly bullied and looked down upon. Gyutaro had a deformed body which added to his misery. He was filled with hatred towards the world that treated them so badly. In order to protect his sister, he became extremely violent and cruel to those who he thought were threats. His experiences in his early life shaped him into the demon he later became, using his powers to lash out at those he despised.
A simple Nezuko and Zenitsu fanfiction could be that they go on a journey together. Along the way, they encounter other demon slayers who are initially skeptical of Nezuko. But Zenitsu stands up for her, explaining that she's different. This shows the depth of their friendship and how Zenitsu has come to trust Nezuko completely.
One common theme is self - discovery. Since Zenitsu has a complex personality, many fanfics focus on him finding out more about himself, his powers, and his place in the world. Another theme is friendship. He often forms strong bonds with other characters in fanfics, like Tanjiro and Inosuke.