Well, he could use the power of friendship. Maybe with the help of his friends like Hermione and Ron, they come up with a plan. They might find evidence that Snape has been involved in some really bad things that go against the wizarding laws. Once they present this evidence to the proper authorities, like Dumbledore or the Ministry of Magic, Snape could be gotten rid of. Also, Harry might have some sort of prophetic dream that gives him the key to getting rid of Snape, just like he had dreams about Voldemort in the original series.
On the other hand, Harry could be proud of himself for standing up for what he believes in. He may think that he has made Hogwarts a better place by getting rid of a teacher who was not treating students fairly. But at the same time, he might have mixed emotions as Snape was also an important part of the Hogwarts environment for so long.
In some versions, Snape uses his knowledge of the Dark Arts and Hogwarts' secrets. He might set up a situation where Umbridge is lured into a restricted area of the castle and gets into trouble with some magical creature or ancient magic, causing her to be removed from her position.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
He uses his knowledge of the castle's secret passages. By using these, he can quickly disappear after a prank. For example, if he turns Snape's quills into snakes during a class, he can run into a passage right behind his seat and be gone before Snape realizes who did it.
In some fanfictions, it could be through powerful magic spells that he discovers or is taught. Maybe he stumbles upon an ancient spell in the restricted section of the library that allows him to summon a family member. Another possibility is that he unknowingly activates some kind of magical artifact that has the power to bring a sister into his life.
In many fanfictions, Harry might use his powerful spells like the Stunning Spell or the Killing Curse if the story allows for such dark magic use. He could also use his intelligence to outwit the dragon, perhaps finding a weakness in its armor or behavior pattern and then using a well - timed spell to defeat it.
Well, in some fanfictions, Harry might use a super - powered Patronus. His Patronus could be stronger than usual, perhaps due to some newfound magic or a special connection he has developed. It could be so powerful that it simply overpowers the dementors and destroys them.
He probably uses magic. For example, to water the plants, he can use a spell instead of a regular hose. And to fence the ranch, he might use a charm to create a magical barrier that can keep unwanted animals or people out.
Maybe he discovers some ancient spells or incantations that don't need a wand. These could be spells that were forgotten over time but are still effective. He spends time researching in old books and uncovers these wand - less magic techniques.
Well, he could use his magic skills secretly. Maybe he casts a spell on the Dursleys to make them docile or distracted long enough to get Harry out. For example, he might use a Confundus Charm on them so they don't notice Harry leaving.