Welch was part of the "Beat movement" of the 1960s and was a relatively known poet. He associated with other Beat writers including Philip Whalen, Jack Kerouac and Gary Snyder. Kerouac wrote about Welch in his novel Big Sur; the character he based off of Welch is called "Dave Wain." Welch had lived with Snyder and Whalen while attending Reed College, where he wrote his thesis on the writer Gertrude Stein.

Welch moved to New York after graduating from Reed College. He began to have emotional problems afterwards and moved to Florida for a course of therapy. He then studied philosophy and English at the University of Chicago before moving to California, where he drove a cab while devoting most of his time to writing. He got involved in the San Francisco literary scene. People who knew him stated he was a very talented writer, but his multiple nervous breakdowns crippled his career.

Welch was depressed and was drinking heavily by 1971. He was last seen on May 23. He was staying with Snyder in Nevada City, California at the time. Welch walked out of Snyder's house, carrying a revolver, and has never been heard from again. He left a note behind staying he had "went southwest," naming his literary executor, and asking Snyder to take money out of Welch's bank account to cover his debts.

Although Welch is presumed to have committed suicide, his remains have never been located.