They could be depicted as having a teacher - student - like relationship in a very twisted sense. Bellatrix, being a powerful dark witch, has knowledge of certain spells and dark magic that Hermione, in her quest for knowledge, wants to understand. But it's not a friendly or positive relationship. Bellatrix might try to mislead Hermione or use her for her own dark purposes, and Hermione has to be careful not to be corrupted.
In some fanfictions, it might be portrayed as a dark and complex relationship. Bellatrix, being a Death Eater, could potentially have Hermione under her control in a way that shows the power dynamic. For example, she might use Hermione for her knowledge or magical abilities while keeping her in a subservient position.
Often, Bellatrix's obsession is shown through her constant pursuit of Hermione. She might be depicted as always trying to get close to Hermione, either to harm her or to somehow possess her skills or knowledge. For example, she could be shown as lurking in the shadows, waiting for the right moment to strike at Hermione.
Their relationship can be shown as a mentorship. Albus, being the more experienced wizard, can guide Minerva in matters of school governance and handling difficult situations. For example, when there's a conflict between different magical factions in the wizarding world that affects Hogwarts, Albus can offer his insights to Minerva on how to keep the school neutral and safe.
Often, Hermione's intelligence is pitted against Bellatrix's madness. The fanfiction might show Hermione trying to outwit Bellatrix while Bellatrix uses her dark magic to keep Hermione under her thumb, like a pet in a way. For example, Bellatrix could be using some dark spell to control Hermione's actions, and Hermione has to use her wits to break free.
There are also fanfictions that take a completely different turn and show a sort of begrudging respect between them. After a series of events where they both learn more about each other's pasts and motives, they start to respect the other's power and tenacity. This might not lead to friendship, but it does create a more nuanced relationship than just pure hatred.
Hermione is often portrayed as the intelligent and resourceful one. She uses her vast knowledge of magic to outwit Bellatrix. Bellatrix, on the other hand, is usually shown as the wild and unpredictable dark witch. Her loyalty to Voldemort makes her a formidable foe for Hermione.
This moment can also lead to a more complex dynamic. Hermione might become more determined to oppose Bellatrix, not just because of their ideological differences but also because of this personal affront. And Bellatrix may see Hermione as more of a worthy opponent or a nuisance that needs to be dealt with more harshly, which would drive more interactions and conflict between them in the fanfiction.
Well, in some fanfictions, it can start as a power play. Bellatrix, being the dark and powerful character she is, uses the physical position as a way to assert dominance over Hermione. But then it can lead to a more complex relationship where Hermione, instead of being completely submissive, challenges Bellatrix, and this can be the start of a strange sort of mutual understanding or even a twisted form of attraction.
Their relationship could be developed through a series of duels and challenges. At first, they see each other as opponents, but as they face off more, they start to understand each other's fighting styles and strategies. This could lead to a grudging respect. They might then start to train together in a way, not to become friends right away, but to better themselves as wizards, with Bellatrix learning some of James's more sportsman - like dueling ethics and James learning about Bellatrix's tenacity in battle.
It could start with hostility. Since Bellatrix was a Death Eater and Hermione was on the side of good, the daughter might inherit some of her mother's prejudices. But as the story progresses, they could find common ground, like a shared interest in magic theory or a mutual friend. This could gradually lead to a more amicable relationship.