Alexander played Little League Baseball and attended Cardinal Newman High School in Santa Rosa, California, where he was named the North Bay League player of the year as a senior. He set the school records for strikeouts in a season and a career and led them to the league championship in 2007. He was drafted by the Cincinnati Reds in the 37th round of the 2007 Major League Baseball Draft but did not sign and instead attended Pepperdine University to play college baseball. He was selected to the all-West Coast Conference Freshman team in 2008 when he was 7–4 with a 4.95 ERA and struck out 106 batters. He was 4–5 with a 4.11 ERA as a sophomore, when he was used as both a starter and a reliever. Between his two seasons at Pepperdine he played for the La Crosse Loggers of the Northwoods League.

Alexander was then drafted by the Kansas City Royals in the sixth round of the 2010 Major League Baseball Draft and signed with them on June 11, 2010 for a $130,000 signing bonus. He made his professional debut that season with the Idaho Falls Chukars of the Pioneer Baseball League, where he was 1–6 with a 5.73 ERA in 12 games (11 starts). He subsequently missed the entire 2011 season due to left shoulder surgery and returned in 2012 to pitch in 10 games (six starts) for the Kane County Cougars of the Midwest League where he had a 2.55 ERA.

Alexander moved between three levels in the Royals farm system in 2013, with five games for the Lexington Legends of the South Atlantic League, 12 for the Wilmington Blue Rocks of the Carolina League and 24 for the North West Arkansas Naturals of the Texas League. Overall, he was 5–1 with a 3.00 ERA and appeared exclusively out of the bullpen. He did not allow a homerun all season and pitched had the second most innings pitched in the minor leagues (75) without a homer. In 2015 he pitched in 35 games for the Naturals and 11 for the Omaha Storm Chasers of the Pacific Coast League. He finished 2–4 with a 4.52 ERA in 67​⁄₃ innings. He pitched for the Gigantes del Cibao of the Dominican Winter League after the season and then returned to Omaha for 2015, where he was 2–3 with a 2.56 ERA in 63​⁄₃ innings over 41 games. The Royals selected him as their Triple-A Pitcher of the Year.

Alexander was called up to the majors for the first time on September 1, 2015 and he made his MLB debut the following day against the Detroit Tigers. He pitched a scoreless ninth inning, retiring two batters on groundouts and then striking out Nicholas Castellanos to end the game. He pitched in six innings over four games for the Royals that season, allowing three runs on five hits with three strikeouts.

Alexander has Type 1 diabetes, a condition that was diagnosed during the 2016 season. He has three brothers, all of whom played baseball. His older brother, Stu was drafted by the Florida Marlins in the 29th round of the 2003 Major League Baseball draft and played in their minor league system until 2009. His younger brother, Jason signed a minor league contract with the Los Angeles Angels in July 2017.

On January 4, 2018, Alexander was traded to the Los Angeles Dodgers in a three team trade that also sent Jake Peter to the Dodgers, Luis Avilán and Joakim Soria to the Chicago White Sox and Trevor Oaks and Erick Mejia to the Royals.