Begin by getting inspired. Look at fan art related to the Lunar Chronicles or discuss the series with other fans to get ideas. Decide if you want to write a story that's a continuation of the original, a prequel, or something completely different. For example, you could write a story about the early days of the cyborg program that created Cinder. When you start writing, keep the tone of the original series in mind. If it's a story full of action and adventure, like the original, make sure your writing reflects that. Also, be sure to stay true to the personalities of the characters as much as possible.
To start writing lunar chronicles fanfiction, begin by choosing a theme or idea that you find interesting within the Lunar Chronicles universe. It could be the complex relationship between the Lunars and the Earthens. Then, think about the characters you'll use. Are you going to stick with the main characters or introduce some new ones? After that, create a setting for your story. Will it be in New Beijing, on Luna, or some other place? Once you have these elements, start writing your first draft, not worrying too much about grammar at first.