Born Ross Smith in Brooklyn, New York, Alexander began his acting career in Broadway productions during the 1920s. By 1926 he was regarded as a promising leading man, with good looks and an easy and charming style and began appearing in more substantial roles. He was signed to a film contract by Paramount Studios but his film debut in The Wiser Sex (1932) was not a success, so he returned to Broadway where his career continued to rise. In 1934 he was signed to another film contract, this time by Warner Brothers Studios.

Alexander was better suited to the Warner Brothers style of film, and the studio persevered with him, gradually increasing the stature of his roles commensurate with his growing popularity with film audiences. His biggest successes of the period were A Midsummer Night's Dream and Captain Blood (1935). During this time Alexander took action to hide his homosexuality and married an actress named Aleta Friele in 1934. The marriage ended the following year when Friele committed suicide.

In his biography of Bette Davis, More Than A Woman, author James Spada writes of Alexander's infatuation with Davis that reached its height in 1936 with a series of love letters sent by Alexander to Davis. Davis, who knew Alexander to be homosexual, at first found the attention amusing and harmless, and did not discourage it. As his attentions continued she found it annoying, and Alexander was eventually confronted by Davis' husband who assaulted him. Alexander soon after married another actress, Anne Nagel with whom he had appeared in the films China Clipper and Here Comes Carter (both 1936 in film|1936), and he attempted to move forward with his career. Warner Brothers had decided by this time that Alexander's potential as an actor was limited, and that his personal problems did not allow him to focus completely on his career. Although they continued casting him in films, the importance of his roles was greatly diminished.

With his professional and personal lives in disarray, and deeply in debt, Alexander shot himself to death at his home in Los Angeles, California, using the same shotgun his wife Aleta Friele had shot herself with two years earlier. His final film Ready, Willing and Able was released posthumously.