To begin with, read the passage through, thoroughly. Read it once more.
Identify who is speaking, and to whom, along with any other characters who may be present. If you can locate exactly where this scene takes place within the play, even better. Consider which events have just occurred - to which they may be reacting, or affected by - and to where the passage is going to lead (which will allow you to look out for dramatic irony and foreshadowing).
Can you identify any prominent themes of the play that are developed in this passage, and if so, how are they advanced/deepened? Equally, how is the main plot or any subplots developed in the passage? Think 'why does this conversation/action take place? Why did the playwright include this scene, with these characters, in this location?' Is action taking place, or is it being conveyed indirectly, e.g. through descriptive speech? What effect does this have on the reader/audience's reaction to the events of the play?
Is the passage in verse, or prose? How does the presence or absence of metre affect the tone of the speech? Are sentences long and flowing, or short and abrupt? Is punctuation abundant or sparse? How do these details betray the mood of the characters or of the scene?
Take a look at the language. Is a particular lexis used that stands out? For example, is there a lot of legal or medical vocabulary? Does this vary between characters, and do they use different dialects? How does this affect your opinion of each character? Are their voices imperative? Questioning? Casual, calm and friendly? How does this affect the mood of the scene, and does it jar with the content of their speech?
Identify any uses of imagery, and analyse each in turn. Why did the playwright use this metaphor? Why did they use this simile? Equally, be aware of phonic effects such as alliteration and the effect that they have on the tone of a passage. Does the musical effect compliment or unhinge the mood of the scene?
Note the stage directions. Are the actors uncomfortably close when speaking, or deliberately distanced, or are they at ease with eachother? Are they moving, or still? Consider how space would be used in a performance of the scene, and how that would affect the audience's reception.
It may be relevant to consider how genre is manipulated or referred to in the scene - consider any appeals to traditionally tragic or comedic forms, and how they are imitated or distorted by the playwright, if present.
Finally, reflect on how your close reading has been influenced by your own interpretation of the passage. Attempt to let go of any preconceptions you may have, and to approach the text from a perspective that is alternate or somehow opposite to your own. Repeat the process, and see what else you can pick up on.