First, Severus would probably try to gain Harry's trust. He would do this by being less harsh on him in class and showing a bit of kindness. Then, he would try to find out more details about the abuse without scaring Harry away. He might enlist the help of some of the more trustworthy students, like Hermione, who could also keep an eye on Harry. Severus would also make sure that Harry has access to any resources he needs at Hogwarts, such as a warm place to stay during holidays if he doesn't want to go back home.
Severus is a very observant character. He may notice that Harry has a lack of certain basic skills or knowledge that a normal - cared - for child would have. For instance, Harry might not know how to handle simple social situations in a healthy way or might be overly defensive. This could lead Severus to suspect that Harry's home environment, where he should have learned these things, is not a healthy one and that he might be suffering abuse.
Well, Severus could be quite sneaky about it. He might start observing Harry more closely in the corridors and classrooms. If he notices signs of abuse, like Harry being overly withdrawn or having unexplained injuries, he could enlist the help of other trustworthy faculty members. He might also try to give Harry some extra tasks or projects in Potions class to keep him occupied and away from his thoughts about his family situation. This way, Severus can gradually build a relationship with Harry and offer him some form of support without making it too obvious at first.
One great fanfiction with this theme is 'A Different Kind of Family' where Severus discovers Harry's abuse and takes him under his wing. It shows Severus's complex character evolving as he protects Harry.
Severus might initially be shocked. He could then feel a sense of protectiveness over Harry. He might start to secretly look out for Harry, like making sure he has enough to eat at Hogwarts and is safe from further harm.
He would become more nurturing. Usually, Severus is seen as cold and aloof, but with Harry in his care, he'd have to show more warmth. He might still be strict, but it would be out of love for Harry's well - being rather than just being his usual harsh self.
One possible plot could be that Severus discovers the extent of Harry's abuse at the Dursleys. He decides to take Harry in out of a sense of duty or perhaps a hidden soft spot for the boy. He then has to teach Harry about magic in a more personal and nurturing way, different from Hogwarts. They might also face opposition from the wizarding community who question Severus' motives.
James might be shocked at first. He could then feel a sense of protectiveness towards Severus. Maybe he starts to see Severus in a new light and tries to find ways to help him, like keeping an eye on those who abuse Severus.
He could start teaching Harry some defensive magic in case the abuser tries to come back. Also, Bill might use his connections to find a healer or a counselor who can help Harry deal with the emotional trauma of the abuse.
One common plotline is Severus becoming Harry's protector. He may start to train Harry privately to make him stronger. Another could be Severus confronting Harry's abusers, whether it's the Dursleys or others in the wizarding world.
The two main characters are obviously Severus Snape and Harry Potter. Severus is the one who adopts Harry, changing from his usually cold and aloof self to a more caring and protective figure. Harry is the abused child who gets a new chance at a better life under Severus's care.