The dramatic monologue, usually used in poetry, expresses the opinion of a character to an audience without the usual back-and-forth dialogue. It provides the poet a chance to express a view from the point of another character. This creates a story while simultaneously persuading the reader to a particular point of view. Sometimes called a "persona poem," this type of poetry shows up in ancient Greek literature but is best known for its use by the Victorian poet Robert Browning
Develop a character and a back story for that character. Write these on a separate piece of paper. You may not use them directly in the monologue, but they'll help direct what the monologue is about and what point of view -- your character's -- it comes from.
Set the theme or purpose of the monologue. This is the thing that you want your audience to understand from it. The theme can range from the serious, such as racism, to the lighthearted, such as the feelings you might have driving for the first time.
Write the monologue, without rhyme, from the point of view of the character. Include a details about the character's background. For example, if the monologue is about poverty and your character is a person of high society, work this into the monologue by showing his viewpoint of poverty.
Use poetic devices, such as alliteration -- repetition of the first syllable among multiple words -- and imagery. These make the monologue different from simple prose, giving it a sense of elegance. Create a rhyming scheme if you want, but poetic monologues do not require this feature.