How did Greek drama and religion influence each other?
Drama was religion in ancient Greece. Rituals began with hymns to Dionysus, the god of wine. Everyone came to sing and dance to insure a good grape harvest; presumably those partaking imbibed as well. At one point, one particular Greek named Thespis began to sing praises to the god on his own, with the others responding -- this was the beginning of the actor and chorus. The ritual became that 50 or so members in the chorus danced and sang in a circle, in the middle of which was the altar to Dionysus. The remainder sat on a nearby hillside and listened -- the audience. Over time these harvest festivities gave way to other themes, and eventually had nothing to do with Dionysus or any god, but began to evolve into what we now know as the classical Greek plays.
First Ancient History, Oxford University Press, 2000. pg. 166
The Ancient Greeks used drama to advance their culture, and in doing so, they promoted the morals of their gods/goddesses system. For example, in Sophocles' Antigone, the playwright uses a clash between brothers, a quest for power, and the idea of inescapable fate to advance the moral that humans must obey the gods and their traditions even if it means physical death.
Practically speaking, if Greek audiences watched these dramas, they were supposed to take away the moral and apply it to their lives. If an invading city or culture tried to force something upon the Greeks that they were against, then the Greeks were expected to fight to the death in defense of their beliefs or traditions.