After transferring from Muskogee High School to Broken Arrow Senior High School before his junior year, Bradley quickly became the starting quarterback on the school's varsity football team and a starting pitcher for the baseball team. After his junior year, Bradley was named to the 2010 Aflac All-American Baseball Classic roster. In 2011, during his senior season, he led the team to a 36–2 record and its first 6A State Championship since 1991. In the championship game against the Rams of Owasso High School, Bradley struck out 14 batters and allowed just two hits over seven innings in a 4–0 win. He finished his senior season with a 12–1 record, allowing just three earned runs in 71 1⁄3 innings, walking 11 batters and striking out 133.
Bradley was expected to be drafted in the first round of the 2011 draft. He committed to play both football and baseball at the University of Oklahoma, but due to his high draft status, eventually signed a professional baseball contract. ESPN baseball insider, Keith Law, ranked Bradley as the number nine prospect in the 2011 draft. Archie was selected as the 7th Overall pick in the 2011 MLB Draft to the Arizona Diamondbacks.
On August 15, 2011, with only minutes left until the signing deadline, Bradley signed a contract with the Diamondbacks for $5 million, to be spread out over five years due to his two-sport abilities. Instead of going to college to play both football and baseball, he was sent to Arizona for workouts at the rookie level before being sent to the Arizona League. Prior to the 2012 season, he was ranked as the 25th-best prospect in baseball by Baseball America.
In 2012, Bradley went 12–6 with a 3.84 ERA for Low-A South Bend (Midwest League). He pitched 136 innings with 152 strikeouts with a WHIP of 1.26.
In 2015, Bradley made the Diamondbacks opening day roster. He made the start in his MLB debut on April 11, 2015 against the Los Angeles Dodgers. Bradley pitched 6 shutout innings, giving up 1 hit and 4 walks, and striking out 6. He also got his first major league hit, a single. Bradley defeated 2014 Cy Young Award winner, and 2014 NL MVP, Clayton Kershaw in the 6-0 Diamondbacks win, becoming the fifth rookie pitcher to make his first start against a Cy Young winner since 2003, and the fourth to win the game. On April 16 Bradley opposed 2014 World Series MVP Madison Bumgarner becoming the first pitcher in MLB history to face the defending Cy Young Award Winner and the World Series MVP in his first two starts. He earned a no decision in 6 2/3 innings pitched in a Diamondbacks 7-6 win in 12 innings.
During a game against the Colorado Rockies on April 28, 2015, Bradley was hit in the face by a line drive off the bat of Carlos González, rendering him unconscious for three minutes and forcing him to leave the game. The following morning, he was placed on the 15-day disabled list with minor damage to his sinus cavity.
After battling for a spot in the rotation, Bradley lost the competition to Patrick Corbin and would serve as a reliever for the 2017 season in Arizona. Sporting a new thick, wild beard, Bradley flourished in the setup role in 2017.
Bradley is a 6'4", 225 lb. right-handed pitcher who features a fastball that sits in the 96-97 mph range, topping out around 100 mph. He also throws a knuckle curveball in the low 80s that he describes having 12–6 movement. He also occasionally throws a changeup.