If Harry Potter was raised by Bellatrix, he would likely grow up in a very dark and twisted environment. Bellatrix is a Death Eater, so Harry would be exposed to the dark arts from a young age. His values and beliefs would be very different from the ones he has in the original story. He might not have the same sense of right and wrong, and could be more inclined towards the use of dark magic.
One possible plot could be Harry growing up believing he is on the right side by following Bellatrix and the Death Eaters. But then he discovers some secrets about his real parents and starts to question everything he's been taught. This leads to an internal struggle as he tries to decide whether to stay with Bellatrix or break away and find his true identity.
Well, for one thing, his view of the wizarding world would be completely different. Growing up with Bellatrix, he'd probably be taught that pure - blood supremacy was the right way. His relationships would also be altered. He might not have the same friends like Ron and Hermione. Instead, he could be surrounded by Death Eaters or those with dark leanings. His magical skills might develop in a more aggressive and unethical direction as Bellatrix would encourage using magic for domination rather than for good.
One possible plot twist could be that Harry, under Bellatrix's influence, becomes a much darker and more ruthless wizard. Bellatrix might train him in the Dark Arts, and he could end up being a sort of anti - hero. He might have a completely different view of the wizarding world and be at odds with his former friends from Hogwarts.
The challenges would be numerous. Socially, he would be an outcast among those who oppose the Dark Arts. His magical education would be limited to Bellatrix's knowledge, which is mainly centered around dark magic. This could put him at a disadvantage when dealing with more complex and positive forms of magic. Emotionally, he would lack the warmth and support that a normal family provides, and this could lead to serious psychological issues.
The story would be drastically different. Harry might not oppose Voldemort. He'd be on the 'dark side' instead. His Hogwarts experience would be non - existent or very different if he even went there. He'd probably use the dark arts more often.
In a Bellatrix and child Harry Potter fanfic, the author might explore the idea of Bellatrix having a change of heart towards Harry. It could start with her seeing some of James Potter in the young Harry. This could lead to a series of events where she tries to protect Harry from other Death Eaters, which would be a huge deviation from her character in the original series. However, it would make for a very interesting and unique fanfic.
One great Bellatrix fanfic is 'The Redemption of Bellatrix'. It shows a different side of her, exploring the possibility of her having a change of heart. It's really interesting to see how the author weaves a new story around her character.
In a time - travel Bellatrix fanfic, she might go back to the time when she was first introduced to the Dark Arts. This could show a different side of her, perhaps a more innocent - looking but still dark - inclined Bellatrix. She could interact with younger versions of other Death Eaters and form different alliances that change the course of the future in the wizarding world.
One great 'Harry Potter x Bellatrix fanfic' is 'The Unlikely Bond'. It explores a very different relationship dynamic between Harry and Bellatrix. It shows how Harry, in this fictional scenario, tries to understand Bellatrix's twisted mind and in the process, they develop a strange connection that is full of tension and mystery.