Well, since Moody is a very vigilant and somewhat paranoid character, Harry would be more observant. He would constantly be on the lookout for danger, like Moody does. His sense of humor might also be a bit darker, influenced by Moody's experiences in the wizarding wars. He might not be as quick to make friends, as he would be more focused on who could be a potential threat. Also, his view of authority might be different. Instead of always looking up to Dumbledore, he might question authority more often, as Moody is a bit of a maverick in his own right.
In this fanfic, a really cool plot point might be how Moody's artificial eye plays a part in Harry's upbringing. Moody could use it to teach Harry about seeing through disguises and hidden threats. Also, Moody's no - nonsense attitude might make Harry a more disciplined wizard. He might not tolerate any sloppiness in Harry's magic, which would lead to Harry being a more precise and powerful wizard in the long run.
Harry would likely be more gruff and less trusting. Growing up with Moody, who was always on guard, Harry would have learned to be more cynical. He might not be as quick to make friends as he was in the original story.
Harry would be more cautious. Moody's paranoia would rub off on him. He'd be constantly on guard, not as quick to trust people like in the original series. For example, when he meets new classmates, he might be sizing them up for potential threats instead of being friendly right away.
He'd probably be more refined. The Delacours are known for their elegance, so Harry might have better manners and a more sophisticated way of speaking.
He'd likely be more withdrawn and edgy. Constant exposure to the creepy elements would make him less trusting and more on - guard all the time.
One interesting Moody fanfiction could be where he mentors Harry in a more in - depth way outside of the classroom. It might explore Moody's past experiences with dark wizards and how he imparts that knowledge to Harry for future battles.
Another good character to pair with Moody might be Sirius Black. Given their both having experience with the dark side of the wizarding world, they could team up in a fanfiction to take on some new threat or solve a long - standing mystery that has to do with the dark wizards.
His personality would probably be more aloof. Since basilisks are solitary creatures, Harry might not be as sociable as in the original story.
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.