Often, it's a very twisted relationship. Since Bellatrix is a fanatical follower of Voldemort and Hermione is a Gryffindor and a Muggle - born, their relationship in fanfiction can be about Bellatrix trying to break Hermione's spirit, perhaps through various forms of psychological torture or coercion.
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.
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.
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.
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.
In many fanfictions, the relationship is developed through shared experiences. Maybe they are forced to work together in a dangerous situation. This makes them rely on each other, and slowly, feelings start to change. Writers also play with the idea of the magical bond between mates, which can cause them to be more empathetic towards one another.
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.
Often, Voldemort and Bellatrix's relationship is shown as one of master and devoted servant. Bellatrix is extremely loyal to Voldemort, almost to the point of worship. She will do anything he asks without question.
Often, it starts with Bellatrix feeling a strange sense of connection or pity for Hermione. Maybe she sees a bit of herself in the abused girl. Then, as she takes Hermione in, there's a period of distrust from Hermione's side. But over time, they start to understand each other more, and Bellatrix might use her knowledge of the dark arts to protect Hermione in ways others can't.
Often, it can be portrayed as an unexpected and somewhat unorthodox relationship. Narcissa, being from a different background and generation, her possessiveness might stem from seeing Hermione's unique qualities. It could be shown as a sort of mentorship - like possessiveness, where Narcissa wants to guide Hermione in the ways of the wizarding world she knows.
Another theme is redemption. Sometimes, the fanfiction might explore the idea of Bellatrix having a change of heart through her interactions with Hermione. It could be that Hermione's goodness starts to chip away at Bellatrix's dark nature.
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.