He has compiled over 600 NHL victories in regular season games. In his 13 full NHL seasons as head coach, he has taken his teams to the playoffs 12 times. In other leagues, he has been head coach of the American Hockey League's Hershey Bears and the Western Hockey League's Regina Pats.
Murray played hockey in his hometown of Shawville growing up, joining the Shawville Pontiacs intermediate club at age 14. He later joined the Rockland Nationals of the Central Junior Hockey League. He attended Macdonald College, a suburban campus of McGill University, which is located in Ste. Anne de Bellevue, Quebec. He returned to Shawville and worked as a gym teacher. He then went into business buying a local motel.
Murray began his coaching career as coach with the Rockland Nationals in 1976 when the team went all the way and won the Centennial Cup of the Canadian Junior Hockey League (CJHL). He earned a good reputation as a coach and was offered a position with the Pembroke Lumber Kings, and then with the Regina Pats of the Western Hockey League (WHL). He left his business interests in the hands of family members, and moved west. He took Regina to the Memorial Cup in 1980. Murray then became the head coach of the American Hockey League's Hershey Bears in 1980–81, and served in that role until he was promoted to head coach of the Washington Capitals, Hershey Bears' parent NHL team, partway through the next season in 1981–82.
In seven full seasons with the Capitals, Murray brought the team to the playoffs each year, and these playoff appearances were the first in franchise history. In his second year, the Capitals won their first playoff series. However, his teams did not advance beyond the second round. He won the Jack Adams Award as coach of the year in 1984. Murray was fired partway through the 1989–90 season, with the team struggling, and was replaced by his brother Terry Murray.
In 1990, Murray became coach and general manager of the Detroit Red Wings. The team had good results in his three seasons, making the playoffs each year, but not getting beyond the second round. Murray remained as general manager in 1993–94 season after the team named Scotty Bowman as head coach. Murray departed the Red Wings following the season.
Murray was next appointed general manager of the expansion Florida Panthers in 1994. In 1996, the young Panthers made it to the Stanley Cup Finals, and Murray was selected as NHL Executive of the Year. He also coached the Panthers for part of the 1997–98 season.
He next joined the Mighty Ducks of Anaheim as head coach for the 2001–02 season. From 2002 to 2004 Murray was general manager of the Mighty Ducks, and again saw his team quickly make a mark in the playoffs, reaching the Stanley Cup Finals. After a disappointing 2003–04 season with the Mighty Ducks, he surprised many by deciding to resign from the general manager post, and became the head coach of the Ottawa Senators.
On February 20, 2007, he became the fifth NHL coach to achieve 600 victories, in a shootout win against the Edmonton Oilers. Despite this impressive number of victories (at the time the most among active NHL coaches), Murray never won a Stanley Cup. In his most recent trip to the Finals as head coach in 2007, the Senators team that he coached lost in five games to his former club, the Anaheim Ducks. That was the only season in Murray's 17 years as an NHL head coach that his team advanced beyond the second playoff round.
With the firing of John Muckler on June 18, 2007, Murray was promoted to general manager of the Senators, while assistant coach John Paddock took over the club's head coaching duties. However, on February 27, 2008, following a 15–2 start, which had briefly put the Senators in first place in the Eastern Conference, Murray fired Paddock after the team struggled through a disastrous January and February. Murray stepped in as interim head coach for the remainder of the 2007–08 season, finishing with a 7–9–2 record, with the team ultimately finishing in seventh place in the Eastern Conference. The Senators were swept in the first round of the playoffs by the Pittsburgh Penguins.
Craig Hartsburg was hired as the new coach of the team in June 2008. After the Senators struggled for most of the 2008–09 season, Murray fired Hartsburg after a 7–4 loss in the Capitals. In 48 games as head coach of the Senators, Hartsburg posted a 17–24–7 record. Cory Clouston, head coach of the AHL's Binghamton Senators, the team's top farm club, was hired as interim head coach, and Clouston was appointed as head coach on a two-year contract following the end of that season.
Murray signed a three-year contract extension as general manager on April 8, 2011. He fired Clouston and two assistant coaches on April 9, 2011, following the Senators' last game of the season. The team had been beset by injuries to key players such as captain Daniel Alfredsson and star forward Jason Spezza, leading to a mid-season collapse. Murray made a flurry of trades in 2011, after the Senators had fallen out of contention, and promoted many younger players from the team's Binghamton farm club.
In 2015, he was inducted to the Ottawa Sports Hall of Fame.
Murray, one of ten children born to Clarence and Rhoda Murray, was born and raised in the small Ottawa Valley town of Shawville, Quebec, near Ottawa. He and his wife, Geri, have two daughters, Heide and Brittany.
His younger brother, Terry Murray, was head coach for the Washington Capitals, Philadelphia Flyers, Florida Panthers and Los Angeles Kings. His nephew, Tim Murray, previously served under him as assistant general manager of the Senators, and later served as the general manager of the Buffalo Sabres.
In July 2014, the Senators website announced Murray was diagnosed with cancer and was undergoing treatment. On November 13, 2014, Murray announced he had stage 4 colon cancer, which had spread to his liver and lungs. He said "there is no cure for me at this point" and that he may have had cancer for up to ten years before its detection.
On August 12, 2017, Murray died of colon cancer at the age of 74, three years after he was diagnosed.