Often, Hermione and Ron might still be part of the story. They could be curious or even a bit skeptical about Harry's new family situation. And Dumbledore might also make an appearance, perhaps guiding Harry through the process of understanding his new family through blood adoption.
Harry himself is obviously a common character. Then there could be the adoptive parents, who might be original characters created by the fanfiction writer or perhaps some minor characters from the original Harry Potter series given a bigger role.
In some 'harry blood adoption fanfiction', there could be other magical creatures or characters related to the family he's adopted into. For instance, if the adoptive family has a connection to centaurs, a centaur character might be introduced to teach Harry about their unique magical knowledge. Or there could be a family house - elf who has a special bond with Harry after the adoption and helps him navigate the new family dynamics.