Greek tragedies are made up of various components that make them unique. Aristotle defined the Greek concept of a tragedy in his work "Poetics." Choose a Greek tragedy such as "Antigone" by Sophocles and analyze it for the elements that make it a tragedy. Some things to look for include the moment of catharsis (defined by Aristotle as the purging of fear and pity), identification of the "tragic hero" and finding the point of peripeteia, the moment that fate suddenly reverses direction.
Choose a portion of one of the great Greek tragedies (such as "The Oresteia" by Aeschylus or Sophocles' "Oedipus the King") and coordinate a production of the work. This could involve choosing one small segment to be presented to a class or putting together an entire play complete with costumes. To truly get into the spirit of the Greeks, you could slate the performance for late March or early spring, which is when tragedies were performed in honor of the god Dionysus.
Decide on one Greek tragedy to transform into a modernized piece, either as a play or literature. Read this work thoroughly, making sure to understand all the plot elements before transforming it. Try to keep the integrity of the themes and characters as true as possible while entirely transporting the setting to a new place and time. Be sure to think deeply on how the problems humanity dealt with more than 2,000 years ago are still very much alive today.
Once you have become thoroughly familiar with the components of Greek tragedies and have read a few, you can compose your own. Create a plot that is single rather than double (meaning that the outcome does not differ depending on a character's moral alignment). To make the plot complex instead of simple, incorporate elements such as peripeteia (reversal) and anagnorisis (recognition). Don't forget to infuse the entire plot with a great deal of pathos, or suffering, which is the trademark of the Greek tragedy.