In Act Two, how does Biff's realization that his life is a lie underline the theme of the play?
One of the themes of Death of a Salesman is clarified by Biff in the Requiem. Based on his new understanding of life, his father, and himself that he has gained over the course of the play, Biff declares that Willy "had the wrong dreams" and that "he never knew who he was." Thus the play's message is that if one lacks a sense of identity, he will spend his life pursuing the wrong dreams.
In the first part of the play, Biff plays along with Hap and Willy, proceeding on the assumption that he had been a top salesman for Bill Oliver, and that Oliver will be happy to see him again and advance him money for a business venture. But in Act Two, Biff has an epiphany after he steals Oliver's fountain pen. He asks himself why he is trying to become something he doesn't want to be, and then says, "All I want is out there, waiting for me the minute I say I know who I am!" When Biff realizes the connection between identity and pursuing the right dream, the dream that will bring true satisfaction, he becomes willing to look at himself objectively. He admits that he was never a salesman for Oliver--he was a shipping clerk. He admits he doesn't have what it takes to make it in business--that he's "a dime a dozen."
With this new understanding, he determines to make his father face the truth about his sons. He says, "The man don't know who we are! The man is gonna know! We never told the truth for ten minutes in this house!" He confronts Hap with his lie: He is not an assistant buyer, but an assistant to the assistant. He then pleads with Willy to burn his "phony dream" of Biff becoming ultra successful. Biff is unable to persuade Hap and Willy to look at their lives realistically and to create different dreams for themselves based on the truth.
After Willy's death, Hap embraces Willy's dream "to come out number-one man." But Biff has changed, saying "I know who I am, kid." This gives hope that Biff will not follow in his father's footsteps, pursuing a dream that is not based on his own true identity and therefore bound to fail. Biff's realization that he has been living a lie and his determination to face the truth underscores the theme that one can never reach satisfaction in life by pursuing a dream that is not grounded in the truth about who one is.
Biff tells Hap that he has come to the conclusion that the Lomans never tell a word of truth in that house. The house itself is built on lies as all the improvements to its structure were stolen materials from other building projects. So, also, is the image of Biff as a person false. Exaggerated bits from others are assembled and augmented, turning the vision of Biff into a well-liked super athlete bound for unfathomable wealth and success. Biff knows instead, that he is a "dollar an hour" and nothing special.
See the theme "Appearance versus Reality"