Well, in this fanfic, Umbridge might adopt Harry to use him as a pawn in her political games within the wizarding world. She could parade him around as a sign of her 'benevolence' while secretly trying to suppress any signs of rebellion in him. Harry, on the other hand, would be constantly looking for ways to escape her clutches and get back to his real friends and the fight against Voldemort.
One possibility is that Umbridge, despite her cruel nature, sees some potential in Harry for her own gain. Maybe she tries to mold him into a perfect little wizard according to her strict and often wrong ideas. She could isolate Harry from his friends and force him to follow her rules blindly, which would create a lot of internal conflict for Harry as he tries to resist her brainwashing while living under her roof.
In a 'Harry Potter adopted by Umbridge' fanfic, it could start with some sort of legal loophole or magical obligation that forces Harry into Umbridge's care. Umbridge would likely impose her extreme views on discipline and blood purity on Harry. She might even try to rewrite his history of fighting against Voldemort to make it seem like she was the one guiding him all along. Harry would have to struggle not only with her daily abuse of power but also with the false image she tries to create of him in the wizarding community.