A thesis statement is merely your answer to a question about the story, so what sorts of questions about perverseness can you ask about the story? What triggered the narrator's perversity--alcohol or something more intrinsic? What does the sort of perversity that would provoke a man to cut out a cat's eye with a pen-knife tells us about the narrator's character? That he's a sociopath? What is the most perverse act of the story and why? Is it his inexplicable attack on the cat, or his cold-blooded murder (and subsequent cover-up) of his wife, and his ability to sleep well afterwards (for the first time in days), merely because the cat had gone missing?
Thank you for your prompt reply but I feel like the questions you have offered are not broad enough to substantiate a whole research paper. Your inquiries seem more like topic sentences or points to present in a chunk.
I would suggest you start with three steps that relate to one another.
First, review what a thesis statement is. You need to have a clear statement that can support an interpretation throughout your essay.
Second, define what you mean by "perverseness." You've already said that you don't want to write about how everyone is perverse in some way. That's a good idea: that's too general and weak an idea to really do this story justice. So, what do you mean? Does perverseness include the way the story is told, with the narrator distorting things to fit his needs? Or is it just the violence and intense emotion the narrator shows in the story?
Third, review the story one piece at a time, honing your definition and testing your thesis. For example, in the very first paragraph, the narrator says the events in the story have "destroyed" him. If that's the case, can we trust anything he says?