Joseph attended Franklin High School in Franklin, Tennessee, where he played for the high school baseball team. He attended Lipscomb University, choosing to play college baseball for the Lipscomb Bisons in the Atlantic Sun Conference, due to the promise of more playing time than he was offered by Vanderbilt University and the University of Arkansas, where he was also recruited. In 2007, he played collegiate summer baseball with the Falmouth Commodores and Cotuit Kettleers of the Cape Cod Baseball League. In his junior year, Joseph led the Bisons with a .342 batting average, 17 home runs, and a .615 slugging percentage. He was named to the All-Conference team and was named the Most Valuable Player of the conference tournament.

Joseph is married with two children. Joseph’s brother, Corban, is also a baseball player. Corban was also selected in the 2008 draft, chosen by the New York Yankees in the fourth round. They played together with the Orioles in 2018. Their father, Mark, played college baseball for Lipscomb University.

The Baltimore Orioles drafted Joseph in the seventh round, with the 206th overall selection, of the 2008 Major League Baseball draft. He signed with the Orioles, receiving a $125,000 signing bonus and tuition payments for his final three semesters of college. In his first professional season, the Orioles assigned him to the Aberdeen IronBirds of the Class A-Short Season New York–Penn League, where he batted .261 with eight home runs and 34 runs batted in in 63 games. In 2009, the Orioles assigned Joseph to the Frederick Keys of the Class A-Advanced Carolina League.

He played for the Bowie Baysox of the Class AA Eastern League in 2010, and for the Scottsdale Scorpions of the Arizona Fall League after the season. The Orioles returned Joseph to Bowie in 2011 and 2012. In 2013, Joseph had a .299 batting average, 22 home runs, and 97 RBI for the Baysox. He was named an All-Star at mid-season and for the season-ending team, and was named the Eastern League’s Minor League Player of the Year by Topps. At the mid-season Eastern League All-Star Game, Joseph won the Home Run Derby. The Orioles did not renew his contract, and he was not selected in the Rule 5 draft in December 2013.

With Matt Wieters still recovering from Tommy John surgery, Joseph began the 2015 season as the Orioles starting catcher. Joseph was hitting .243 with 4 HR and 18 RBI in 43 games until Wieters was activated on June 5, and following his activation, Joseph and Wieters split catching duties. In 100 games in 2015, Joseph hit .234/.299/.394 with 11 HR and 49 RBI. Joseph also caught 33% of attempted base stealers, and led the pitching staff to a 3.65 ERA when he was behind the plate.

After driving in a pair of runs in a 14–8 home victory over the Kansas City Royals on September 11, 2015, Joseph would not be credited with a run batted in (RBI) in any MLB regular season game until 19 months and 18 days later. The span included the entire 2016 campaign in which he had 132 at-bats and 141 plate appearances, breaking a major-league record which was previously held by Wilbur Wood who had 96 and 124 respectively in 1971. The drought ended when he hit a two-run homer off Tommy Layne in the ninth inning of a 12–4 away loss to the Yankees on April 29, 2017.

In February 2017, the Baltimore Orioles beat Joseph in arbitration, and he was given the team’s $700,000 offer (an increase from his prior salary of $523,500), instead of his $1 million request.

Joseph struggled early in the 2018 season. On May 18, the Orioles optioned Joseph to Norfolk.

On February 13, 2019, Joseph agreed to a one-year contract with the Arizona Diamondbacks. On December 2, 2019, Joseph was non-tendered by Arizona and became a free agent.

On January 19, 2020, Joseph signed a minor league contract with the Toronto Blue Jays. On September 5, the Blue Jays selected Joseph from their taxi squad, then he was designated for assignment on September 11. Joseph elected free agency on October 15, 2020.