Analyzing a play is not different from analyzing a novel or a poem. Structure must be considered and theme, literary devices, plot, conflict, symbolism and imagery, setting and time (a distinct part of setting). There are some features particular to plays that must be included in an analysis (Vonier College). One of these is whether the action is mental or physical action. One might say Death of a Salesman is a play with mental action as is Faust Part II whereas King Lear and Faust Part I might be said to have physical action. Another of these particular features is whether there are soliloquies and/or monologues. Another is characters since characters have a singularly significant function in plays. The final feature particular to analysis of plays is key lines.
To apply some of these points to Shaw's The Man of Destiny, it is a one act play with four characters, two of whom are the primary characters. It is based on an historic incident at the early stage of Napoleon Bonaparte's military career following upon his advancement to General. As such, Bonaparte is the lead character and an unnamed Lady is the second principle character. They meet at an inn and the inn keeper, Giuseppe, is instrumental in helping to reveal information about them to each other and to the audience. The fourth character, the Lieutenant, is instrumental in setting up the conflict, which is introduced at the very beginning of the play and pursued hotly from the start. The action is mental; nothing physical rally takes place except for the innkeeper's comings and goings. There are no soliloquies in this play, but Shaw does give Bonaparte a monologue (i.e., long speech by one character that interrupts conversation) that reveals his philosophy and point of view when he, a Frenchman, elaborates his theory of the national character of the English and of the English moral conscience (this is in a sense ironic since Shaw is himself an Englishman).
The plot is a simple one: a Lady has stolen Bonaparte's letters from his Lieutenant on the highway in order to remove a personal one written by Josephine to Director Barras and maliciously sent to Bonaparte. He has encountered her and attempts to retrieve his mail while she attempts to keep him from reading the contents of Josephine's personal letter. The conflict of the mental action then is the battle between them for possession of the physical letters and then the contents of the one letter. The theme is that of how destiny turns on single events and remarks. Two key lines are spoken by the Lady because her brief comments, in the face of sure defeat, turn events in favor of her achieving her end by raising the right questions and motives in Bonaparte's mind. The lines are:
LADY. Nothing— (He interrupts her with an exclamation of satisfaction. She proceeds quietly) except that you will cut a very foolish figure in the eyes of France.
LADY (springing up with a bright flush in her cheeks). Oh, you are too bad. Keep your letters. Read the story of your own dishonor in them; and much good may they do you. Good-bye. (She goes indignantly towards the inner door.)
The other elements for analyzing The Man of Destiny you can easily discover through your own reading of the play while watching for symbolism, imagery, setting, time, and other literary devices such as techniques of irony or metonymy and structural elements like climax and resolution.