Murray was born and raised in Thunder Bay, Ontario and played his minor hockey with the Thunder Bay Minor Hockey Association. He spent his younger years playing AA before moving up to the Thunder Bay Kings AAA program.
In his minor midget season (2009–10), Murray led his Kings to a 3-2-0 record at the OHL Cup. That season with the Kings, Murray appeared in 40 games with a 2.28 GAA and 6 shutouts. He was subsequently selected in the 2nd round (35th overall) of the 2010 OHL Priority Selection by the Sault Ste. Marie Greyhounds.
Murray played major junior hockey in the Ontario Hockey League with the Sault Ste. Marie Greyhounds. On September 4, 2013, Murray was signed to a three-year entry-level contract with the Pittsburgh Penguins.
During his first full professional season in 2014–15, on March 8, 2015, Murray set an AHL record for the longest shutout streak by not allowing any goals for 304 minutes and 11 seconds. The previous record of 268:17 was held by Barry Brust.
On March 22, Murray recorded his tenth shutout of the season, breaking a record for rookie AHL goaltenders set by Gordie Bell in 1942–43. He would finish the regular season with 12 shutouts. Murray capped his standout rookie season by sweeping the AHL awards, he was selected to the AHL First All-Star Team and Rookie Team, while winning the Baz Bastien Memorial Award as the League's best goaltender and the Dudley "Red" Garrett Memorial Award as best rookie.
In the 2015–16 season, Murray was recalled from the AHL for the first time to the Pittsburgh Penguins on December 15, 2015 and made his NHL debut on December 19, 2015 in a 2–1 loss against the Carolina Hurricanes. Murray was recalled again by the Pittsburgh Penguins on February 21, 2016 and started nine games to finish the season.
On April 19, 2016 the 21-year old Murray made his NHL playoff debut, allowing just 1 goal as the Penguins defeated the New York Rangers 3–1. Two days later, on April 21, Murray recorded his first NHL playoff shutout in a 5–0 win over the Rangers.
Murray remained the starting goaltender throughout the playoffs upon replacing Jeff Zatkoff in game 3 of the first round. He was replaced by Marc-André Fleury just once after returning as the starter. He started all but one game of the remainder of the 2016 Stanley Cup Playoffs, including every game of the 2016 Stanley Cup Final. He became only the sixth starting rookie goaltender in the Cup Finals since 1976.
To start the 16-17 season, Murray was injured due to a broken hand sustained during international play. Following his return, he supplanted the veteran Fleury as the Penguins starter, and retained his elite form from his first year. During the season, he recorded 32 wins, a .923 save percentage, 2.41 GAA, and 4 shutouts. He finished within the top 10 of the NHL amongst qualifying goaltenders in wins, save percentage, and shutouts, while finishing 11th in GAA. This elite play allowed Murray to finish 4th in Calder Trophy voting for Rookie of the Year, and earned Murray a spot on the 2016-17 All-Rookie Team and the 2017 NHL All-Playoff Team,
He recorded his first NHL point on December 8, 2016, an assist on an empty net goal.
During warm-ups of Game 1 of the Penguins' first-round matchup versus the Columbus Blue Jackets in the 2017 Stanley Cup playoffs, Murray was injured and Fleury took starting position. He saw his first action of the 2017 Stanley Cup Playoffs in Game 3 of the Eastern Conference Finals against the Ottawa Senators after Fleury was pulled. He recorded a shutout two games later, stopping all 25 shots he faced in a 7-0 win. A thrilling Game 7 OT win helped Murray and the Penguins reach the Stanley Cup Finals for the second year in a row. Despite two solid starts in Game 1 and 2 against the Nashville Predators, Murray's game faltered in Games 3 and 4. As a result, there were speculations on who would start in Game 5. Murray was announced as the starter and stopped all 24 shots the Nashville Predators threw at him in a 6-0 win. Murray made 27 saves in Game 6, shutting out the Predators 2-0 and winning the Stanley Cup for the second consecutive season.
Murray went on to represent Team North America at the 2016 World Cup of Hockey, as starting goalie. On September 19, he broke his hand playing against Russia but went undiagnosed for several days before getting an MRI and confirmed with a broken ligament.
Murray's father is from Scotland and his mother is from the Netherlands.