In that fanfiction, it could be that a character who was secretly working for Voldemort, perhaps someone we didn't suspect at first, like a seemingly innocent teacher or a student who was under the Imperius Curse. Another option is that a group of dark wizards invaded Hogwarts and Harry was unfortunately caught in the crossfire and killed. Or perhaps a magical artifact malfunctioned and ended up being the cause of his death.
In the fifth Harry Potter book, Sirius Black dies. Voldemort succeeded in luring Harry and his companions to the Department of Mysteries because Kreacher, the house-elf, had revealed to the Death Eaters that Harry regarded Sirius as his father. Sirius Black and the Order of the Phoenix went to the rescue and fought fiercely with the Death Eaters in the Department of Mysteries 'death chambers. Sirius Black was struck by Bellatrix Lestrange's Death Curse and died. Harry Potter and the Goblet of Fire was not enough. Everyone, please click to read the novel!
The characters who died in the seventh Harry Potter movie (both in the movie and in the original novel) included Hedwig (killed by the Reaper Curse), Mad-Eye Moody (killed by Voldemort), and Dobby was killed by Bellatrix in the Deathly Hallows (Dobby died in the movie Deathly Hallows Part 1). In addition, many characters died in the final battle, such as Lupin and Tonks. Harry Potter and the Sorcerer's Stone was not enough. Everyone, please click to read the novel!
It could mean that the popularity and strict control of the 'Harry Potter' franchise somehow stifled the creation of fanfiction. Maybe the official regulations or the overwhelming dominance of the official story made it difficult for fanfiction writers to explore and create freely.
The relationships that Harry was supposed to form later in life, like his friendships with Ron and Hermione, would never happen. And the whole wizarding world would be different. Without Harry, there might not be someone to stand up to Voldemort in the same way. So, the power balance in the wizarding world would be completely shifted. Also, the mystery around Harry's scar and his connection to Voldemort would be left unresolved in a way that leaves the story with a sense of incompleteness.
Quirrell could be the main villain. He was already under Voldemort's influence in the original story. It's possible that in this fanfiction, he doesn't hold back and attacks Harry more directly and lethally during his first year. Maybe he uses his position as a teacher to get close to Harry and then strikes when he has the chance.
If Harry Potter died at the Dursleys, it would be a huge shock to the magical community. First, the Dursleys, who had always resented Harry, would suddenly find themselves in a very difficult situation. They were supposed to protect Harry, but his death would surely draw the attention of the Ministry of Magic. In fanfiction, this could lead to investigations into the Dursleys' treatment of Harry. Magically, the protection that Harry's mother's sacrifice provided might be broken in a different way. And Voldemort, unhindered by Harry, could start to reshape the wizarding world according to his dark desires much more quickly.
On the other hand, there are fans who appreciate the creativity of such fanfictions. They see it as a new take on the familiar story, exploring different themes and emotions that come with the death of the main character. It can also lead to interesting discussions among fans about the nature of the Harry Potter universe and how it could be different if he had died.
Perhaps in the fanfiction, Harry Potter's death occurs because he tries to rewrite a powerful magic spell that goes horribly wrong. He was always known for his bravery and willingness to take risks in the original series, so this could be an extension of that. His death would then be a tragic consequence of his own curiosity and desire to push the boundaries of magic.
I don't think it's common. The Harry Potter fanfiction community usually focuses on things like exploring different relationships between characters, or imagining new adventures. 'Hermione killed fanfiction' is quite an odd and not - so - often - used phrase in that context.
The person who killed Barty Crouch in fanfiction is highly unpredictable. It could be a dark wizard who emerges from the shadows or a close ally with a hidden agenda.