Born in Phoenix, Arizona, Graham began writing comedy in high school, selling material to humorist Robert Orben for seven dollars a joke. While still in college, he wrote for Mad magazine, creating scripts for artist Don Martin.
Soon after graduation, Graham was hired as a writer on the sitcom Benson.
After this, he worked as a writer and producer on the last three seasons of the long-running ABC comedy, Who's the Boss?. During this period, Graham wrote "The All-Nighter," an episode still somewhat controversial among fans of the sitcom because of its relatively dark themes.
In 1995, Graham became the head writer and an executive producer of The Drew Carey Show. He remained on the show for eight of its nine seasons, producing more than 200 episodes. He produced and helped write many of the series' trademark musical numbers, as well as its annual "live" broadcasts. In 1998, Graham and Drew Carey took a small production crew overseas and became the first Americans to film a sitcom in China, using locations such as The Great Wall and the Beijing McDonald's.
In April 2009, TBS announced at its upfront that it was developing a prime-time, animated comedy series with Graham called "Big Tow." 
The New York Times published an essay by Graham in its Modern Love column on July 8, 2010 titled "What Clown Wrote This Script?' detailing Graham's relationship with an unnamed actress for whom he developed a TV pilot a decade earlier. Although comic in tone, the essay explores Graham's inner struggles about mixing business and romance and ends on a bittersweet note.
In 2007, Graham began developing MrsP.com, a new entertainment website for children. Co-created with Kathy Kinney, who played "Mimi" on The Drew Carey Show and who also stars as "Mrs. P" in the site, and Dana Plautz, a former executive with Hanna-Barbera and Intel, the comic website celebrates reading and launched on November 10, 2008.
- Newsweek, Nov. 16, 1998. Newsmakers
- http://www.MrsP.com. "About us"