Research Woody Allen films. Make it your business to consume all Woody Allen material. Pay special attention to those films that have gotten critical acclaim such as "Annie Hall", "Bullets over Broadway", "Hannah and Her Sisters", "Crimes and Misdemeanors" and "Husbands and Wives".
Write down your initial impressions of the work even if you have seen them before. Try to record your first reactions. Determine what in the material inspired you to laugh, cry, think or be disturbed. Notice how your reactions are carefully crafted by Allen's writing and story development.
Find a sense of neuroses that is present in your life; Woody Allen is famous for his neurotic characters that seem fearful or anxious in the face of everyday life yet still manage to get into extraordinary situations. Draw inspiration from your own fears and phobias as Allen does and use those to create a basic backdrop for a story idea.
Write short stories to help you gain a sense of your own personal style. Write stories that make you feel a specific way and invoke a certain mood. Inject a fair amount of humor, since it will make your writing most reminiscent of Allen's. Juxtapose even the most dark of moments in the screenplay with a sort of existential wit that helps the reader to stay engaged and a bit uncomfortable with the material.
Research philosophical models like existentialism, nihilism and reductionism to get a backdrop or idea of how Woody Allen uses these ideas in his own work. Consider building an intricate and involved story line only to break it down to mean nothing at the end of the story or screenplay to capture some of Allen's signature techniques.