Well, Klaus and Dave's relationship in the fanfiction can be quite different from the show sometimes. Some fanfictions might focus on their backstories together, like how they first met. Others could explore what might have happened if Dave had survived. It really depends on the author's take, but generally, it's a relationship full of love and the struggle to be together given all the obstacles in the Umbrella Academy universe.
Another theme is his relationship with his family. His siblings may not fully understand his pain, and this can lead to situations where he gets hurt, either by their actions or inactions. His family's neglect or misunderstanding of his power - related issues can be a central part of the hurt in these stories.
Their powers are very different, Diego with his ability to manipulate knives and Klaus with his ability to communicate with the dead. This contrast creates an interesting dynamic in fanfiction as they can complement each other's skills. For example, Klaus might get information from the dead that helps Diego in a fight.