Well, Percy would be more attuned to the feminine energy around him. Artemis and her hunters are all female, so he would be more sensitive to the emotions and needs of women. His sense of humor might also change. Instead of the typical sarcastic and boyish humor, he could develop a more dry and understated humor. His view of heroism would also shift. Instead of the traditional heroics of fighting and winning battles, he would see heroism in protecting nature, the weak, and upholding the values of the hunt. He would be a more complex character, with a blend of his original self and the new traits instilled by Artemis.
Percy's personality would be more in tune with nature. Since Artemis has a strong connection to nature, Percy would grow up respecting and understanding it more deeply. He would probably be more solitary as well, as the hunters often keep to themselves. He might not be as quick to trust others as he used to be, having been brought up in a more exclusive group. Also, he would have a greater sense of loyalty towards Artemis and the hunt, which could make him seem a bit cold to outsiders.
Percy could be more in tune with his emotions. Dionysus is associated with passion and ecstasy, so Percy might be more openly emotional, not suppressing his feelings as much as he did in the original stories. He could also be more creative, as Dionysus has an element of creativity in his domain. For example, he might be better at coming up with unique solutions to problems using his more artistic side.
He'd likely be more reserved. Artemis values solitude and purity, so Percy might be less of a joker. He'd be more serious about his duties and tasks.
His communication style could be different too. He might be more direct and simple in his speech, like how animals communicate basic needs and emotions. He wouldn't be as interested in the complex social hierarchies and politics among demigods. Instead, he'd focus on the more straightforward relationships based on mutual respect and survival, much like the relationships in the animal kingdom.
He'd probably be more brooding. Growing up with Hades, he'd be exposed to the darker side of the mythological world constantly. So, instead of the somewhat upbeat Percy we know, he might be more withdrawn and moody.
He might be more feral. Growing up with a monster, he could lack some of the social graces that a normal upbringing in the wizarding world would provide. He may be more direct and less concerned with propriety.
He might be more laid - back. Canadian culture is often seen as more relaxed, so Harry could have a calmer demeanor. He wouldn't be as quick to get into arguments or be as impulsive as he sometimes was in the original story.
Harry might be more confident in his pure - blood heritage. Since Dorea Black would likely educate him about it, he wouldn't feel as out of place among the pure - blood families as he did in the original story. He may carry himself with more pride, but not in an arrogant way.
If Percy is the brother of Artemis and Apollo, his personality might be more refined. Since Artemis is associated with grace and Apollo with beauty and culture, Percy could be more sophisticated in his mannerisms compared to the original Percy. He might be more attuned to the arts and have a deeper sense of propriety.
His sense of humor would likely be different. Sirius has a bit of a mischievous streak, and Harry would pick up on that. He might be more prone to making jokes and playing pranks in a more light - hearted way compared to the somewhat more reserved Harry we know from the original story. Also, Harry would be more trusting of others in general, especially those in the wizarding world, as Sirius would introduce him to a more positive side of it from the start.
He'd be more in tune with the sea. He might be calmer, like the ebb and flow of the ocean. His reactions to things would be more measured as he's used to the slow, powerful nature of the sea. For example, instead of getting angry quickly, he'd wait and think like the sea does before a big wave hits.