The essential element of drama is "conflict", by which is meant the story to unfold is the battle between two forces of roughly equal strength, in a closed space (so that the pressure and tension canot be relieved by separating the two combatants), with rising complications leading to a climax and resolution. (Much of this vocabulary is borrowed from Eugene Scribe's amalysis of the well-made play structure.) So your analysis of a drama/play can start with a close look at what the conflict consists of--opposing social forces? Opposing values (perhaps from two generations)? Opposing loyalties? etc. As for character analysis,the same kind of scrutiny will serve--does the character ring true, is the character consistent, is the character suffering from an internal conflict of values? etc.