Born in Los Angeles, California, Greene learned how to pitch at the Major League Baseball Urban Youth Academy in Compton.

Greene suffered an ulnar collateral ligament injury partway through the 2018 season and underwent Tommy John surgery the following year.

[4][5] At Notre Dame High School in Sherman Oaks, Los Angeles, Greene played shortstop when he was not pitching.

[2] Over four high school baseball seasons, Greene had a career 1.62 earned run average (ERA) in 121+1⁄3 innings pitched, striking out over 30 percent of the batters he faced.

[13] Greene, who had been committed to play college baseball for the UCLA Bruins since he was a freshman in high school,[14] ultimately agreed to a professional contract with the Reds only a few minutes before the 2 p.m. (PDT) signing deadline on July 7.

Once he signed with the Reds, Greene was assigned to the Billings Mustangs, their farm system team in the Rookie-level Pioneer League.

[19] The Reds medical staff had hoped that the injury would improve through nonsurgical rehabilitation, but in March 2019, Greene suffered a setback, and he underwent Tommy John surgery to repair the ligament.

[22] He made seven starts there, during which he went 5–0 with a 1.98 ERA and struck out 60 batters in 41 innings, before receiving a promotion to the Triple-A Louisville Bats on June 15.

[25] After impressing coaches during spring training, Greene made the Reds' Opening Day roster for the 2022 MLB season.

[26] He made his major league debut on April 10, earning the win in a 6–3 Cincinnati victory over defending World Series champions the Atlanta Braves.

[27] Facing the Los Angeles Dodgers on April 17 for his second start, Greene set an MLB record by throwing 39 pitches at speeds of 100 mph (160 km/h) or higher.

It was his third outing in which he pitched at least six innings, struck out eight or more batters while allowing no more than one hit, a record for rookie pitchers in the live-ball era.

[32] He was transferred to the 60–day injured list on July 31,[33] and began a rehab assignment with the Arizona Complex League Reds the following day.

[37] The year after his silver medal performance at the U15 tournament, Greene once again represented the United States at the 2015 U-18 Baseball World Cup in Japan.