Harry being raised by Walburga Black would have a profound impact on his personality. Since Walburga was a firm believer in pure - blood supremacy and had a rather tyrannical nature, Harry would grow up in an environment full of rules and expectations based on blood status. He might become more arrogant towards those not of pure - blood heritage. His sense of humor, which was a big part of his canon personality, might be stifled. He could be more focused on upholding the family's 'proud' pure - blood name and less interested in making friends with Muggle - borns or half - bloods. However, as he got older, he might also start to rebel against the values he was raised with, much like Sirius did against his family.
On the other hand, he could become more powerful and confident in his magic earlier. Walburga might push him hard in his magical studies, and with her knowledge of the Black family's magical heritage, Harry could develop a more assertive and perhaps even a bit of an arrogant streak in his magical abilities. But at the same time, he would also have to deal with the internal conflict of the Black family's dark past.
His life would be quite different. Walburga was a strict and proud pure - blood. Harry might grow up with a stronger sense of pure - blood superiority drilled into him. He could be more proficient in dark arts - adjacent magic earlier as Walburga was from the Black family which had some connections to such magic. However, he would also likely face more internal conflict as the values of Hogwarts and his friends would clash with what Walburga taught him.
Harry would likely be more confident. Sirius is a bold and brave character, and growing up with him would instill those qualities in Harry more strongly. He wouldn't be as meek as he was in the early days at Hogwarts.
Harry would probably be more aloof. The Black family has a certain air of aristocracy. So, he might be less approachable than in the original story. He could also be more secretive, having grown up in a family where family secrets are highly guarded.
His relationship with Sirius would be extremely strained. Since Walburga was Sirius' mother and they had a very difficult relationship, Sirius would likely see Harry as a product of his mother's influence. Harry, on the other hand, might be initially hostile towards Sirius due to being raised with Walburga's views.
If Harry was raised by Walburga Black, his relationship with Hogwarts would be drastically altered. Firstly, he might enter Hogwarts with a preconceived notion of his superiority due to his pure - blood upbringing. He could look down on some of the traditions that were inclusive of all blood types. His loyalty to Hogwarts as a whole might be divided. He would likely be more drawn to the Slytherin ideals which are more in line with Walburga's beliefs. He might not form the same close - knit group of friends from different houses. Instead, he could be more focused on building relationships with pure - blood students from Slytherin or other houses who adhered to similar values. His view of Hogwarts as a place of unity and acceptance would be clouded by the pure - blood ideology he was raised with.
He'd likely be more fearless. Dragons are fear - inspiring creatures, and growing up with them would make Harry less intimidated by threats. He might be more direct in his actions and less likely to second - guess himself.
Harry might be more secretive. He'd have to hide any doubts or conflicting feelings he has about the dark path he's being led down. Bellatrix would not tolerate any signs of weakness or deviation from her plans. So, he'd become very good at hiding his true self, which would make his personality complex and full of contradictions.
He might be more shrewd. Goblins are known for their business acumen and cunning. So, Harry could be more calculating in his decisions, not as impulsive as in the original story.
His personality would likely be more reserved. Growing up with the Greengrass family, he might not be as impulsive as in the original story. He would be taught to think before acting, a trait often associated with Slytherin. So, he would be less likely to jump into dangerous situations without a plan.