Claudio attended Isabel Flores High School in Juncos, Puerto Rico. Claudio was drafted by the Texas Rangers in the 27th round of the 2010 MLB draft. Claudio spent his first professional season of 2010 playing for the AZL Rangers of the Rookie-levelArizona League, going 0–1 with a 6.60 ERA in 15 innings. He split the 2011 season Between the AZL Rangers and the Spokane Indians of the Class A Short SeasonNorthwest League, going a combined 5–0 with a 1.91 ERA over 28 innings. He returned to the AZL Rangers in 2012, going 4–0 with a 1.79 ERA over 45 innings. Following the 2012 season, Claudio converted from a traditional over the top pitching delivery to a sidearm delivery. Claudio played for the Hickory Crawdads of the Class ASouth Atlantic League and the Frisco RoughRiders of the Double-ATexas League in 2013, going a combined 4–6 with a 1.83 ERA over innings. He split the 2014 minor league season between the Myrtle Beach Pelicans of the Class A-AdvancedCarolina League, Frisco, and the Round Rock Express of the Triple-APacific Coast League, combining to go 6–3 with a 1.66 ERA over innings. The Rangers selected Claudio's contract and promoted him to the major leagues for the first time on August 13, 2014. He pitched a scoreless inning, and struck out Kevin Kiermaier for his first major league strikeout in his debut that night. He went 0–0 with a 2.92 ERA over 12 innings for Texas in 2014. Claudio split the 2015 season between Texas and Round Rock. With the Rangers, he went 1–1 with a 2.87 ERA over 15.2 innings. With Round Rock, he went 3–1 with a 2.93 ERA over 40 innings. Claudio once again split the season between Texas and Round Rock in 2016. With the Rangers, he went 4–1 with a 2.79 ERA in innings. With Round Rock, he went 0–0 with a 0.55 ERA in 16 innings. Claudio was named the 2017 Texas Rangers Pitcher of the Year, following a season where he posted a 4–1 record with a 2.50 ERA, 56 strikeouts, and 11 saves over innings in 70 games. In 2018, Claudio went 4–2 with a 4.48 ERA over 68 innings. Right-handed batters had a higher batting average against him,.359, than against all other MLB pitchers in 30 or more innings.
Milwaukee Brewers
Claudio was traded to the Milwaukee Brewers on December 13, 2018, in exchange for a competitive balance Round A draft pick. In 2019, he posted a 2–2 record with a 4.06 ERA over 62 innings. He led all major league pitchers in games played. Claudio was non-tendered and became a free agent on December 2, 2019. He re-signed with Milwaukee on a one-year contract on December 9, 2019.
