Read the monologue over and over. By rereading you will be able to familiarize yourself with the words and phrasing before you start memorizing.
Find the rhythm. Shakespeare wrote in many forms, but most of his writing was done in iambic pentameter. This means it is an unstressed word or syllable followed by a stressed word or syllable. Iambic pentameter has a certain rhythm and just like memorizing a song, if you get the rhythm in your mind it can help you to memorize a Shakespearean monologue.
Break them up. Not necessarily by line, but by line of thought. Each line can be only part of a sentence, so if you want to break up the monologue into sizeable chunks that involve one train of thought or one subject, it can be easier to digest.
Consider the monologue's context in the entire play. By getting the idea of what the story plot line is (by reading it, watching the movie or seeing the play performed live), you can better understand what the monologue is about. Keeping the words in context can help you to memorize a Shakespearean monologue.
Keep the words alive. Even though the speech can be a bit boring and long, keep the words alive by sticking to each word and its pronunciation. You can't really ad lib with Shakespeare, so be thorough and adamant about getting each word right in speech.
Practice, practice, practice. There is nothing like long periods of time when it comes to memorizing anything. If you take the time to devote to the task, it will be the best thing you can do to ensure that you can memorize a Shakespearean monologue.