Born in Pasadena, Texas, into a third-generation deaf family, Harvard is the younger of two deaf sons of Kay (Youngblood) and Henry Harvard. Both his parents and his paternal grandmother are also deaf. In the early 1980s, the Harvards moved to Austin, Texas so that their elder son Renny could enroll at their alma mater, Texas School for the Deaf (TSD). The family initially placed Russell (due to his speech capability and residual hearing) in an oral school for children who learn to lip read exclusively. Finding he was unhappy there, his parents switched him to a deaf school education at TSD, which included training in lip reading and speech therapy in English. Although he is able to hear some sound with the use of a hearing aid, including speech and music, he identifies himself as deaf and considers American Sign Language to be his first language.

After graduating from TSD in 1999, Harvard began his studies at Gallaudet University in Washington, D.C. At various times during his college education he took a hiatus to work as a teacher's assistant for preschoolers at the Alaska State School for the Deaf and Hard of Hearing in Anchorage, Alaska. (His mother later joined him, working for the American Red Cross.) While there he contemplated a career as a Teacher of theater, and in 2008 he returned as Artist in Residence. At Gallaudet he maintained a high GPA and completed his bachelor's degree in Theatre Arts, graduating in 2008.

Harvard cites his seeing, at age eight, his cousin perform on stage in The Wizard of Oz as the inspiration for his becoming an actor. Subsequently, he became very involved in theater at TSD. At Gallaudet he appeared in a 2006 stage production of A Streetcar Named Desire and as Claudio in their co-production with Amaryllis Theatre of Much Ado About Nothing. His earliest professional stage work was in the twin roles of the Orderly and the Groundskeeper's Son in the world premiere of Rachel Sheinkin and GrooveLily's Sleeping Beauty Wakes for Deaf West Theatre (Center Theatre Group, 2007). With this dual performance, wrote TimeOut critic James Sims, "Harvard joins the rank of deaf actors transcending any perceived limitations due to a lack of speaking lines, capturing the heart of the newly created characters with ease." The following year, he played (also for Deaf West) Aesop in Aesop Who? In 2007, he assistant directed the young-audience musical Nobody's Perfect (Kennedy Center and VSA Arts).

On May 26, 2010, Harvard testified as a panel witness before the U.S. Senate Subcommittee on Communications, Technology and the Internet in a hearing entitled "Innovation and Inclusion: The Americans with Disabilities Act at 20", which focused on the issues raised by the Equal Access to Communications in the 21st Century Act. Speaking on behalf of the National Association of the Deaf and the Coalition of Organizations for Accessible Technology (COAT), he argued the need for Congress to pass new legislation to require closed captioning on video programming on the Internet just as it had done for television broadcasting in 1990:

In August 2013, the FX/MGM production team of Fargo, the anthology TV miniseries adaptation of the 1996 Coen brothers' film, cast Harvard as Mr. Wrench, one of two hitmen who pursue Billy Bob Thornton's lead character Lorne Malvo throughout the series. Writer-creator Noah Hawley, a part-time Austin resident who lives near Texas School for the Deaf, cited his own neighborhood encounters with sign language as the inspiration for the "Mr. Wrench" character: a deaf Assassin who uses his command of ASL as a means of menace toward his targets and of private communication with his partner Mr. Numbers (played by Adam Goldberg). During the five-month shoot in Calgary, Alberta, Harvard and the show's ASL manager, Catherine MacKinnon, worked closely with Goldberg on translating the pair's dialogue into the most effective ASL exchanges for their scenes.

Critical response to Harvard and Goldberg's seriocomic turn as bickering hired killers was overwhelmingly positive. Reviewers noted that they "steal scenes as Mr. Numbers and Mr. Wrench" (Time), make up one of the "satisfying subplots" (Huffington Post) and "have their own original energy" (Vulture.com). For Alan Sepinwall of HitFix.com, "the relationship between Goldberg . . . and Harvard feels unlike any Criminal twosome of its type I've seen before, even in the midst of a show that is otherwise cleverly rearranging familiar pieces of the movie and other crime stories." And Tim Goodman, TV critic for The Hollywood Reporter, wrote, "Encapsulating everything that is joyously weird about Fargo, the killers are the dangerous—and deaf—Mr. Wrench (Russell Harvard) and his partner and translator, Mr. Numbers (Adam Goldberg) . . . Already I want a separate series that just follows around Mr. Wrench and Mr. Numbers." Series creator Hawley, moreover, who has described Harvard as "magnetic and charismatic" in the role of Mr. Wrench, ended up extending the character's appearance in the series. On June 19, 2014, the Broadcast Television Journalists Association honored Fargo with three awards (including Best Mini-series) at the Critics' Choice Television Awards ceremony. Fargo also won three Emmys—most prominently "Outstanding Miniseries"—at the 66th Annual Primetime Emmy Awards on August 25, 2014; "Best Mini-Series or Motion Picture Made for Television" at the 72nd Annual Golden Globe Awards on January 11, 2015; and, for "majestically reinventing a beloved tale and for expanding and richly rendering a darkly comic world of crime, revenge, and comeuppance", was honored with a 2014 Peabody Award, whose citation recognized Fargo as having set "a new standard for what is possible in the process of adaptation."

On August 10, 2015, Playbill.com officially announced the casting of Harvard for the Broadway company of Deaf West's revival production of the Duncan Sheik-Steven Sater musical Spring Awakening. This transfer production, which opened at the Brooks Atkinson Theatre on September 27, 2015, marked the Broadway debuts of both Harvard and his co-star Marlee Matlin.