Find an appropriate monologue for you. This is the most important step, and where less judicious performers will fly off the rails. You have to pick a monologue that suits you. Your looks, age, sex, vocal quality and skill level need to be taken into account when choosing a monologue to help you avoid choosing one that just won't work for you. For instance, no matter how much work you may do, you won't be a convincing Richard III or Willie Loman if you're 19 years old with a boyish face, nor will you be able to fully realize the depths of Hamlet's "To be or not to be..." if you're in your first acting class.
Analyze the piece. Emotional contrasts and bold choices in your performance won't come from divine inspiration. To fully embrace the monologue, you must analyze the text until you understand every word of what's being said by your character. Basic questions to ask and answer about your chosen monologue are: Who is the character speaking to, and what are the circumstances? What does the character want? Why does he want it? How is he going about getting it? Some of these questions are clearly open to interpretation and will have more than one right answer, and that's okay, as long as you have your own answers based on your understanding of the text.
Memorize your monologue. The importance of this can't be overstated. Nothing will ruin the flow of your performance like having to stop to recall the next line. Committing your monologue to memory will allow you to recite it effortlessly and naturally. If you truly take time commit it to memory, you'll find you can recite it almost unconsciously, which allows you to think about your character's circumstances and create internal dialogue to enrich your performance. The easiest way to learn a monologue is to break it into chunks. You can commit two sentences to memory much more quickly and thoroughly than three paragraphs.