Death of a Salesman is unique inthat, even though it is a play, it still uses internal focalization, meaning that it directly and indirectly presents events as they are seen by the constrained eye of a specific character. This makes it feasible to consider that just about any character in the play as capable of telling a story of their lives.
However, consider that Biff may very well be the one, truly, round character in the play. He is the only one who steps out of the "dysfunctional dance" that Willy Loman has created for his family. He has come as far as to question his life, the origins of his thoughts, and the constructs that he had once built for himself. The fact that he wishes for a change tells a lot about his ability to analyze and ponder upon things. Hence, Biff is the best candidate for keeping a diary.
If he were to write an entry, the topic that is most likely to surface is how the fall of his father from the pedestal of admiration that Biff had built for him has, to this day, affected his life. Finding out that his once-thought almighty and flawless father is a false idol is what ultimately turns Biff's life upside-down, causing him to lose faith in everything, from fate to life, to himself. This introspective topic, so unique to Biff, is what makes him change, turning his character from static to dynamic.