Learn the pronunciation of the letters. The Greek alphabet consists of both upper- and lower-case versions of its letters. Find a chart of the letters and their names in transliteration on Omniglot.com (see References 1). To learn the Greek alphabet fast, break the series of letters into sections of five. Learn these and memorize them. This will help your process go faster and you will feel more accomplished than if you try to memorize the whole alphabet at once.
Use flashcards to test your memory of the letters. At first, when learning from the chart, you can learn the upper- and lower-case together so that you can associate them with one another. However, when you start using flashcards to memorize the letters, make a separate card for upper- and lower-case letters. This will help you be sure that you are confident with using both versions of the letters on their own when writing or spelling.
Practice writing. You must learn to write the alphabet, not just be able to recognize it. Practice this as you make your flashcards. Once you have memorized and can write all the letters, make flashcards for some words as well. Making the flashcards can be part of the study process as you learn some new vocabulary. Use some of the words in English that have been borrowed from the Greek, as they will be more familiar to you. (See References 3.)
Review the alphabet and some basic words you have learned. Keep what you have learned fresh in your mind. Put the names of the letters to the "ABC" song and sing it to yourself, silently and out loud. Have a friend help you with your memorization by dictating the names of the letters or spelling out a word for you as you listen and write it down.