Weather is often used to set the tone in fiction writing. For the writer, it's a very convenient tool—a way to inject descriptive, sensory notes into the writing that make the reader feel more immersed. You'll find examples in many genres of fiction, but there are a few areas that may be especially useful to you as you complete this project:
First, you might consider taking a look at nautical stories. At sea, the stakes of the weather are uniquely high. Because a bad storm may very literally be life or death, weather is often a narrative focus. In classic works like Melville's Moby-Dick, Coleridge's "The Rime of the Ancient Mariner," or Avi's young adult novel The True Confessions of Charlotte Doyle, violent storms often foreshadow (and sometimes directly contribute to) violence and conflict in the characters' lives.
For an example from another perspective, Kazuo Ishiguro's recent novel Klara and the Sun might also be a good one to consider. The story's protagonist, Klara, is an optimistic humanoid robot who runs on solar power. She worships the sun as a benevolent deity, and her unique relationship with the weather is one of the story's major elements.
For yet another angle, you might want to explore speculative fiction. In dystopian stories, extreme weather is often used to connote climate change as a result of human overconsumption, natural disasters, or other forms of societal collapse. Works like Margaret Atwood's Oryx and Crake, Claire Vaye Watkins's Gold Fame Citrus, and Cormac McCarthy's The Road are all excellent archetypal examples here. Jenny Offill's Weather deals, in the present day, with a woman's fear of a similar apocalypse that hasn't actually happened yet. And though the "weather" in this case is technically an airborne industrial disaster and thus manmade, DeLillo's White Noise fits squarely in this category of climate anxiety, too.