How does Sartre use conflict to explore ideas of relationship and thoughts?
The main conflict in No Exit (and in Sartre's philosophy of existentialism) is human vs. human concerning freedom. Remember, No Exit was written during the Nazi occupation of France, so Sartre wanted to show that "hell is other people." In other words, we let other people limit our freedom. But, rather than military occupation, we invite others to torture us. Many of us even enjoy it.
Sartre has three categories of freedom: the human whom he compares to stones, the human he compares to plants, and the true person. The human who is a stone, or who feeds on earthly bread instead of spiritual nourishment, makes no choices and is happy. Sartre believes most people are in this category: they are trapped by social conventions, fear, and false self-image.
Inez, Garcin, and Estelle were like stones when they were alive. Even in the play, after death, they act like stones. They cannot even leave the room they hate (none of them exit the open door at the end). Instead, they exhibit bad faith, a false self-image presented to others. In other words, they wear masks (hide their authentic selves) and play-act (pretend to be someone they are not).
Their relationships with each other is a triangle of anguish (fear of freedom), for each character can foster freedom while the other one blocks it. Inez can love Estelle, but Garcin blocks it. Estelle can free Garcin, but Inez spites him. Inez can free Garcin, but Estelle interferes. And so on. Here is the recipe for absurdity (a cycle of non-freedom).
Much of what hell is for Sartre though his play is conflict in relationships. This is what passes for pure torment. In creating an emotional setting whereby each of the individuals in hell must need the other in order to validate their own sense of consciousness, conflict is inevitable and relationships are predicated on a bizarre dependence under the guise of independence. The conflict in the play arises from the dependence each individual has to another. Inez needs Estelle, who needs Garcin, who needs Inez. This bizarre triangle of mutually assured destruction creates an emotional dynamic where conflict is inevitable. There is little communicated that indicates human beings will ever find a state of totality in such interactions. This might be why they are in hell and why torment is a necessity for each of them. In the end, relationships seem to be predicated on conflict and avoiding it seems impossible.