Well, obviously Spider - Man himself, Peter Parker. He's the main character, and all the stories revolve around him. But other popular characters are his friends like Harry Osborn. Harry has a complex relationship with Peter, and their friendship can be explored in many ways, especially considering Harry's family issues with his father being the Green Goblin. Another popular character is Mary Jane. She can be a love interest, a source of support, or even get involved in some of Spider - Man's adventures.
Definitely Doctor Octopus. He's a classic Spider - Man villain. His mechanical arms and his brilliant but twisted mind make him a great addition to any fanfiction. You could write about his elaborate schemes to take over the city or his begrudging respect for Spider - Man's abilities. And don't forget about Gwen Stacy. She was an important part of Peter's life in different versions of the story. Her relationship with Peter can be explored in fanfiction, whether it's a romantic one or just a friendship that was cut short in the original comics.