This sounds like common sense, but in order for your audience to understand it, you need to understand it first. Your first step should be a thorough reading of whichever piece you intend to perform.
Try to get your hands on a copy with footnotes that will help you understand certain words. Once you understand what all the words mean, interpret the meanings of phrases and write your own notes. Keep an eye out for double entendres and jokes that you can capitalize on in comedies, or especially touching lines in tragedies.
Read the monologue aloud just as you would if you were reading a passage written in today's English. This will help you figure out how things fit together.
As you continue to rehearse and memorize the monologue, you will notice the mood changes and the comedic and dramatic nuances. You will need to emphasize these because it's your responsibility to interpret and send the message to your audience that something is funny or sad. A good strategy is to take it way over the top, then bring it back just a little bit.
Watch Shakespearean performances and notice how the actors bring these elements into their performances.
Practice, practice, practice!