I know Peck drank himself to death in How I Learned to Drive; however, did he fall down stairs or shoot himself while drinking too much?

In How I Learned to Drive, by Paula Vogel (who, by the way, won a Pulitzer Prize for Drama for this work), Li'l Bit is a youngster who has been repeatedly molested by her Uncle Peck (a recovering alcoholic) from a young age.

There are two reasons that "set the stage" for this to occur.  First, Peck and his wife, Li'l Bit's aunt, are among the adults living in the family home.

Second, Peck and Li'l Bit are both outsiders.  Peck has married into the family. Li'l Bit feels like an outsider because she receives no support from her family, specifically about her dream for college; when her mother suspects that Peck's interests in Li'l Bit aren't purely platonic--but she still allows them to be together unsupervised; and, even when her breasts become the open topic of discussion around the dinner table.  (In this already unhealthy environment, her uncle promises that he will not have sex with her until she says it's okay.)

Perhaps because of her uncle's behavior, Li'l Bit is not very well adjusted.  In college she drinks a lot, but the drinking started in the company of her uncle when she was 16.  (Ultimately, she is expelled from college.)

When she turns 18, Peck shows up with the intention of taking their relationship to the next level, now that she is of legal age to give her consent--he even plans to divorce his wife to marry Li'l Bit.  However, Li'l Bit confronts him and tells him that she is breaking off the relationship and does not want to see him again.

Li'l Bit now reveals that at that point, Peck starts drinking, and it takes him seven years to drink himself to death.  This conveys the sense that his death is not a sudden occurrence, but one that takes place over years of alcoholic abuse.  It is the drinking itself that ultimately kills Peck.  One could argue that this long period of "self-abuse" seems an apt punishment for a predator like Peck.