Disclaimer: These are only suggestions and you should let you creativity soar!
Know your world. It helps to develop as much context as you can for your monologue. Who is your character? What are the important relationships in their life? How old are they? Where are they right now? What time period is it? What external factors inform your character’s internal conflict?
Motivate your monologue. Characters will likely have a reason why they give a monologue. What was the conflict or event that prompted them to speak? Are they speaking to another character? Are they convincing someone? Telling a story? Asking a question? Reflecting on the past or experiencing the present?
This can change throughout the course of the monologue.
Find your character’s voice. Everybody has a particular way that they speak. A 70-year-old retired British professor sounds very different than a bratty toddler. How does your character sound? What kind of language do they use? Do they speak in long or short sentences? Do they second-guess themselves a lot or do they plan out what they’re going to say?
Don’t be afraid to tell a story. Monologues can still absolutely have a clear beginning, middle, and end. For the beginning, try to grab the audience’s attention while also giving context for why the character is speaking. For the middle, try to move the “plot” along while also providing plenty of detail and small twists. For the end, try to find a way to wrap up the character’s thoughts without over-explaining. Trust the audience to derive meaning from the monologue themselves.
Delete half of what you write. This may sound a bit extreme, but you can often tell the same story in half of the words you originally do. What can be inferred in between the lines? How can you “show” something happening versus “telling” it?
Read it out loud. Monologues are meant to be performed. Try having a friend or family member read your monologue out loud to you and really listen to how it sounds. Does it tell the story you want it to? Does it seem natural? Does it feel too short or too long? This will likely inform your revision process.