A new character could be introduced. Maybe an old wizard who has been living in isolation for years and has a completely different view of magic compared to Hogwarts. This wizard could take Harry under his wing and teach him things like ancient spells and forgotten magical arts that are not taught at Hogwarts.
Ron could also be important. He might be torn between his loyalty to Hogwarts and his friendship with Harry. Eventually, he could choose to leave Hogwarts to be with Harry, bringing his Weasley family charm and his own brand of magic to the group. The Weasley twins might also get involved, providing some much - needed humor and their unique magical inventions.
The main antagonists could be those who don't want Harry to receive a proper Hogwarts education. Maybe some dark wizards who fear that Harry will become too powerful if he goes through the Hogwarts training. So, they plot to keep him away from Hogwarts.
One possibility could be that Harry discovers he has some other magical destiny outside of Hogwarts. Maybe he is taken in by a different group of wizards who teach him their own brand of magic in a hidden location. They might believe Hogwarts is too mainstream or that Harry has a special power that needs to be nurtured away from the school.
Viktor Krum could be a major character. Since he is from Durmstrang, he might play a role like a mentor or a rival to Harry. Another character could be the headmaster of Durmstrang. He would be important as he sets the rules and tone for the school.
The Dumbledore character could still be important. He might be secretly watching over Harry, guiding him from a distance. Another character could be a new wizard or witch that Harry meets in the Muggle world who has a special mission related to Harry. For example, a young witch who has been exiled from the wizarding world but has the knowledge to help Harry understand his powers. And of course, Voldemort would still be a main antagonist as his threat looms over the entire wizarding world whether Harry is at Hogwarts or not.
There could be a strict drill sergeant named Sergeant Thompson. He is tough on Harry at first but later comes to respect him. Another character could be Harry's roommate, Jack. Jack is a typical military school student who helps Harry understand the ropes.
There could be a French wizard named Pierre who becomes Harry's friend and helps him adjust to the new school.
One possible plotline could be that Harry discovers a hidden magical world outside of Hogwarts. Maybe he stumbles upon a group of rogue wizards who teach him their own brand of magic, away from the traditional Hogwarts curriculum.
Obviously, Harry would be one of the main characters. He is the central figure in the Hogwarts world. Hermione and Ron could also play significant roles as they are Harry's closest friends and are often involved in his adventures.
Another character could be a cologne - making elf. In the wizarding world, there could be elves who specialize in making colognes. This elf could have a special bond with Harry. It could be that the elf was tasked with making a special cologne for Harry's protection from the moment he was born. And as Harry grows, the elf helps him understand the true power of the colognes he has been surrounded with all his life.
Hermione could be one. She might follow Harry to America to continue their adventures together. She could help Harry adjust to the new place with her intelligence and knowledge.