In certain 'Harry kills the dragon' fanfictions, Harry could be given a special magical weapon specifically designed to slay dragons. He might have to go through a series of trials to figure out how to use this weapon effectively. It could be that the weapon has to be powered by his own magic and emotions. Once he masters this, he is able to take on the dragon and kill it. For example, the weapon might need to be charged with his feelings of courage and determination, which are strong in Harry, and then he uses it to strike a fatal blow to the dragon.
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.
It could be an entirely new type of dragon created specifically for this fanfiction. The author might have wanted to introduce a unique dragon with special abilities, like a dragon that can change its color to blend in with its surroundings or one that can breathe not just fire but also some kind of magical ice. Harry would then have an even greater challenge when trying to kill it with the sword.
One consequence could be that he becomes a hero among his peers. Everyone in Hogwarts might look up to him even more. The other students could see him as a real dragon - slayer, which gives him more popularity and respect.
In many Harry Potter - related stories, the sword of Gryffindor often finds its way to those who are worthy. Maybe in this fanfiction, Harry discovers it in a similar way. It could be hidden in the Sorting Hat, or perhaps it's presented to him by some magical creature that recognizes his need and his courage.
Well, it could be that in this fanfic, Harry has some sort of special artifact or weapon. For example, he might find an ancient magical object that has the power to vanquish dementors. He figures out how to use it and thus is able to kill the dementors. It could also be that he has made a special connection with some magical being or force that gives him the strength to take on the dementors in a more lethal way.
He may use a powerful spell combination. For example, he could start with a Confundus Charm to confuse the Horntail and then follow up with a blasting spell to deal a significant amount of damage.
It could be in a fanfic where Harry teams up with someone powerful or uses a clever strategy that no one expected. The story could also change the nature of the task or the dragon itself to make it possible for Harry to succeed early on.
Well, in some fanfictions, Harry might use some hidden magic he discovers in Azkaban. Maybe he finds a way to turn the Dementors' power against Voldemort. Since Dementors can suck out a person's soul, Harry could somehow direct them towards Voldemort, weakening him enough to deliver the final blow.
In many fanfictions, Percy might use his water powers in a unique way. Maybe he combines his control over water with his sword fighting skills. He could create a giant water vortex to disorient Kronos and then strike with Riptide while Kronos is off - balance.
He probably feeds it a diet of magical creatures. Dragons might like to eat things like hippogriff chicks or giant spiders.