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.
One great Klaus 'Umbrella Academy' fanfiction is 'The Ghostly Encounters of Klaus'. It delves deeper into Klaus's relationship with the spirits he can see and communicate with. It adds more emotional depth to his character.
There's also a fanfic called 'Klaus and the Mysterious Artifact'. In this story, Klaus stumbles upon an ancient artifact that seems to have a strange connection to his family's past. It's full of adventure and mystery, and the author does a great job of writing Klaus' character as both humorous and complex.