In "The Crucible," how do the performers demonstrate the tensions in their relationships through their actions?
In the film version of The Crucible, that the tension between John Proctor and Abigail Williams is demonstrated through their eyes. How she looks at him is very telling, it is obvious that she pines for him, loves him, longs for him. The tension of having to stay away from her has made John Proctor act cautiously around her, staying at an arm's distance when they talk alone together in Act I.
In Act II, in the Proctor home, the tension between husband and wife is evident in her wooden actions of serving the evening meal. There is little conversation, and it is very formal. Elizabeth has an unforgiving nature, her face is stern and cold.
Act III the tension is obvious in the posture of Nurse, Proctor and Corey. These practical men appeal to the court on behalf of their jailed wives, believing that the court is interested in the truth. When the truth is treated with disregard and twisted into evidence against innocent people, the outrage is evident in the tension expressed by the pleading of these men, who stand in great disbelief at the court's decision.
Act IV, the tension is at its highest point. The respectable members of society, Corey, Nurse and Proctor are scheduled for execution. Hale is so full of tension that he can barely contain his speech, he becomes shrill with hysteria in his attempt to save the lives of the innocent.
I think what you're asking about is the way in which an actor reveals subtext. It's one of the more difficult parts of crafting a performance, but also one of the most rewarding. It's especially compelling in a play as layered as Miller's "Crucible". Because so many of the characters are restrained in their speech, because of what other characters are present or because of the social mores of the time, a lot of the communication must come nonverbally.
Actors need to carefully consider the ways in which they use their physicality - when they make eye contact, when they look away, the direction their body inclines (towards or away the other characters), the location of their personal tension (in hands, for example. They also have their vocal instrument - thinking about inflection, pitch, tempo, etc.