Frank David Greene grew up in San Jose, California, in a non-musical family. He began studying trumpet with local teacher Bill Nicolosi, who first exposed him to the playing of Maynard Ferguson, and while still in high school he was already performing with Dave Eshelman's San Jose City College Big Band. After graduating he enrolled at the University of North Texas (then North Texas State University), where he played lead trumpet in the school's famed One O'Clock Lab Band, was awarded the Marvin Stamm Scholarship, and studied with brass faculty John Haynie and Don Jacoby. Before finishing his degree he joined Woody Herman's Thundering Herd, then spent the next five years as lead trumpet, recording and touring with Maynard Ferguson.

Greene relocated to New York City after leaving Ferguson's band and quickly became one of the city's most in-demand lead trumpet players, a role he has held for four decades. His theater and studio work has included lead, solo and section trumpet on Broadway for "The Producers" (with Matthew Broderick and Nathan Lane) and its film soundtrack, Disney's "The Little Mermaid" on Broadway, and the 2023 New Broadway cast recording of "Funny Girl." His big-band career reads as a survey of the postwar lead-trumpet chair: the Dizzy Gillespie Alumni/All-Star Big Band under Slide Hampton, the Clark Terry Big Band, the Roy Hargrove Big Band, the McCoy Tyner Big Band, the Frank Foster Big Band, the Jimmy Heath Big Band, the Frank Wess Nonet, the Jon Faddis Big Band (formerly the Carnegie Hall Jazz Band), the Gerald Wilson Orchestra, the Maria Schneider Orchestra, the Bob Mintzer Big Band, the Vanguard Jazz Orchestra (formerly the Thad Jones/Mel Lewis Jazz Orchestra), the Christian McBride Big Band, Mike LeDonne's Big Band, Chuck Owen and The Jazz Surge, and the Count Basie Orchestra.

Among that long list of section credits is a 2006 recording session at Avatar Studios in New York for Anat Cohen's large-ensemble album Noir, released the following year on Anzic Records as "Anat Cohen & The Anzic Orchestra." Greene played trumpet and flugelhorn in the orchestra's three-piece trumpet section alongside Cohen's brother Avishai Cohen (trumpet) and Tanya Darby, arranged and conducted throughout by Oded Lev-Ari. He also recurs in Jimmy Heath's working big band across more than a decade, from the studio album Turn Up the Heath (2006) through the live document Togetherness (recorded 2011, released 2013), where the trumpet section's five chairs included Roy Hargrove.

Since August 2012, Greene has also held the trumpet chair in Paul Shaffer and the World's Most Dangerous Band, the house band Shaffer led for "Late Show with David Letterman" as the CBS Orchestra through the show's 2015 finale and has continued to lead in occasional reunions since. Greene is also the author of a lead-trumpet method, "The Quiet Mind," and an accompanying instructional series, "Brass Concepts," drawing on his years of Broadway pit and big-band lead work.

Greene built his career on the specialized, high-pressure discipline of big-band lead trumpet: setting a section's phrasing, dynamics and time feel from the first chair rather than soloing as a featured voice. He has spoken about tailoring his equipment (mouthpiece rims and backbores) individually to each chair he holds, using different setups for the Dizzy Gillespie tribute band, the Jimmy Heath and Frank Foster big bands, the Roy Hargrove Big Band, and the Village Vanguard Jazz Orchestra. He cites Maynard Ferguson and Don Jacoby's shared philosophy that a trumpeter's job is to project a compelling sound the audience can hold onto rather than perform technique for its own sake, and names Al Porcino, Snooky Young, John Audino, Conrad Gozzo, Arnie Chycoski, Bill Chase, Earl Gardner, Jon Faddis, Byron Stripling and Wayne Bergeron as the lead-trumpet lineage he studied.