Hammond played his junior hockey in the British Columbia Hockey League (BCHL). He played one game with the Westside Warriors in the 2006–07 season before being cut by the team and returning to the Grandview Steelers of the Junior "B" Pacific Junior Hockey League to finish the season. Hammond played 32 games with the Surrey Eagles to start the 2007–08 BCHL season before being traded to the Vernon Vipers for $2,500. In his final season with the Vipers, the team won the Royal Bank Cup national Junior "A" championship and Hammond posted a 2–0 shutout in the final game.
Hammond attended Bowling Green State University where he played four seasons (2009 to 2013) with the Falcons ice hockey team. In 111 NCAA games, he earned a record of 30–68–13 and was named Bowling Green's Most Valuable Player for both the 2010–11 and 2011–12 seasons. It was at Bowling Green that Hammond's ability to "steal" wins for the Falcons prompted teammate Wade Finegan to start calling the goaltender "Robber," then "Burglar," then simply "Burgs." The nickname finally evolved to "The Hamburglar."
The NHL's Ottawa Senators signed Hammond as a free agent goaltender to a two-year, entry-level contract on March 20, 2013. He was subsequently assigned to the team's American Hockey League (AHL) affiliate, the Binghamton Senators, where he finished the 2012–13 season.
Hammond attended the Senators' training camp ahead of the 2013–14 season, but was later designated to Binghamton on September 30, 2013. He was recalled by Ottawa on February 26, 2014, to replace goaltender Craig Anderson, who was with his wife for the birth of the couple's second child. The next day, Hammond, who backed-up goaltender Robin Lehner, entered a game against the Detroit Red Wings at 5:05 of the second period after Lehner gave up six goals and was pulled. Hammond stopped all 11 shots he faced in the remainder of the game.
Hammond attended Ottawa's training camp ahead of the 2014–15 season, but on September 29, 2014, was again designated to Binghamton to start the season. He made his first career NHL start against the Montreal Canadiens at the Canadian Tire Centre on February 18, 2015, after goaltender Robin Lehner was injured in Ottawa's previous game. Hammond earned his first NHL victory in the game, stopping 42 shots in a 4–2 victory, also being named the game's First Star. Two games later, he recorded his first career NHL shutout as the Senators defeated the Anaheim Ducks 3–0, and the next day, he recorded his second consecutive shutout in a 1–0 win over the Los Angeles Kings. Hammond's four consecutive wins marked only the second time of the season that Ottawa won more than two games in a row. On March 2, 2015, Hammond was rewarded for his surprising success as the NHL named him the First Star of the Week. After winning a game against the Carolina Hurricanes in overtime on March 17, Hammond became just the second goaltender in NHL history to allow two goals or fewer in their first 12 starts, a feat matching Hockey Hall of Famer Frank Brimsek's record set during the 1938–39 season with the Boston Bruins.
On April 1, 2015, Hammond was named the NHL's First Star of the Month for March after compiling a 10–1–1 record with a 2.09 goals against average (GAA) and a .930 save percentage, helping the Senators climb to within three points of the final Wild Card spot in the Eastern Conference. On the same day, the Ottawa Chapter of the Professional Hockey Writers' Association announced Hammond as Ottawa's 2015 nominee for the Bill Masterton Memorial Trophy, awarded annually to the NHL player who best exemplifies the qualities of perseverance, sportsmanship and dedication to hockey.
Hammond finished the 2014–15 regular season with a record of 20–1–2, helping the Senators clinch a Stanley Cup playoff berth, where they were eventually ousted by the Montreal Canadiens in the Eastern Conference Quarterfinals. Hammond started Ottawa's first two games of the series before being replaced by Craig Anderson for the following four games.
On May 20, shortly after the end of the Senators' season, Hammond signed a three-year, $4.05 million contract extension with Ottawa, carrying an average annual value of $1.35 million.
On February 11, 2017, Hammond's 29th birthday, he was put on waivers by the Ottawa Senators after goaltender Craig Anderson returned from tending to his wife, who was suffering from cancer, and the recent success of backup goaltender Mike Condon.
Hammond was born in White Rock, British Columbia to Marshall and Sandie Hammond, and grew up in adjacent Surrey. He has a brother, Matt, and a sister, Christine.
Hammond earned the nickname "The Hamburglar" during his university playing career at Bowling Green University. A play on his name, the nickname refers to the McDonald's restaurants' mascot character whose likeness, in the guise of MAD magazines' Alfred E. Neuman, Hammond had painted onto his goalie mask.
Hammond has said that he almost quit hockey on two separate occasions, the first when he was cut after a junior A game in the 2006–07 season, and the second when he was traded from the BCHL's Surrey Eagles mid-season in 2007–08.