Clevenger was raised in Glen Burnie, Maryland, attended Overlook Elementary and graduated from Mount Saint Joseph High School in 2004. He was the starting shortstop on the school's varsity baseball team which won the Maryland Interscholastic Athletic Association (MIAA) A conference championship in his senior year.
He hit .347 as a first-team All-Southland Conference shortstop in his only year at Southeastern Louisiana University in 2005. After transferring to Chipola College, he led the Indians with a .395 batting average and 77 hits.
He was selected by the Chicago Cubs in the seventh round (209th overall) of the 2006 MLB Draft. After signing with the Cubs on June 14, he began his professional career as the starting second baseman with the Boise Hawks. He started making the transition to catcher the following season after Hawks manager Steve McFarland convinced him that a left-handed batter at the position had a better chance of a faster promotion to the majors.
During the 2011 season Clevenger played for the Tennessee Smokies where he batted .295 and hit 5 home runs. He also played for the Iowa Cubs where in 86 at bats he batted .407 and hit 3 home runs.
Clevenger was called up to the Cubs from the expanded 40-man roster. He made his major league debut September 26, 2011, however he only played in two games during the 2011 season.
Clevenger made the opening day roster during the 2012 season. However, on April 28, Clevenger was placed on the 15-day disabled list with a strained right oblique. Clevenger finished the 2012 season with a .201 batting average, one home run, and 16 RBIs.
On July 2, 2013, Clevenger was traded to the Baltimore Orioles with Scott Feldman for pitchers Jake Arrieta and Pedro Strop. He was optioned to the Triple-A Norfolk Tides after the trade. He was recalled on September 1, but did not play for the Orioles until September 13 against the Toronto Blue Jays. On August 17, 2015 he became the first Baltimore native to hit a home run for the Orioles at Camden Yards.
Clevenger batted .225/.289/.337 for the 2014 season. Showalter was dissatisfied with Clevenger's defense (he caught 3 of 20 attempted base-stealers), and the team had problems with his conditioning. Clevenger as of December 2014 had one minor league option remaining. Clevenger was optioned to Triple-A Norfolk Tides on April 7, 2015 and on April 24, was placed on the DL with a left thumb contusion. He was recalled to Baltimore on May 26, and made his season debut two days later. He went 2-for-4 with an RBI in the game. Clevenger played in four games before being optioned back to Norfolk on June 5. Clevenger would end up making the IL All-Star team. After several roster moves by the Orioles, Clevenger was once again recalled by Baltimore on August 14. Two days later, he set a New career-high in hits, going 4-for-6 with a double and an RBI in an 18-2 rout over the Oakland Athletics.
On December 2, 2015, the Orioles traded Clevenger to the Seattle Mariners for Mark Trumbo and C. J. Riefenhauser. Clevenger had a poor 2016, appearing in only 20 games at catcher for the Mariners before sustaining a broken hand on June 29 in a game against the Pittsburgh Pirates, an injury that proved season-ending. On November 2, 2016, Clevenger was outrighted to the minors. He elected free agency on November 4.
On July 21, 2017, Clevenger signed with the Lancaster Barnstormers of the Atlantic League of Professional Baseball.
On September 22, 2016; Clevenger tweeted that he thought it was ironic that black people were beating white people after a black man had been shot dead by a black policeman and that Black Lives Matter protestors, who had been destroying property, as well as President Obama and all Black Lives Matter supporters, should be "locked behind bars like animals," prompting an official response from Mariners GM Jerry Dipoto disavowing those remarks. The next day, Dipoto announced that Clevenger was suspended without pay for the remainder of the season.