Murray was born and raised in Kitchener, Ontario, the son of Sylvia and Roger Murray, who was born in Jamaica and moved to Canada at age nine.    He also has a younger brother, Lamar. His father grew up running track and field and playing basketball; as a youth, his father played against Kitchener native Lennox Lewis before Lewis began his professional boxing career. 
When Murray was three years old, he could play basketball "for hours" and played in a league for ten-year-olds when he was six.   By the age of 12 or 13, he began playing pick-up games against top high school and college players. His father put him through many basketball drills and kung fu exercises, including meditation.  
High school career
Murray attended Grand River Collegiate Institute in Kitchener, later transferring to Orangeville Prep in Orangeville, Ontario, where his father served as an assistant coach.  He and fellow prospect Thon Maker formed a duo that would help Orangeville Prep defeat many American schools. 
On June 24, 2015, Murray committed to Kentucky to play for coach John Calipari.    Murray is the only Kentucky player to have scored over 30 points in more than one game during a freshman season. He is currently tied with Terrence Jones for most points in a game by a UK freshman (35). He was featured on the Midseason Top 25 list for the John R. Wooden Award on January 13, 2016.  He was also named to the 35-man midseason watchlist for the Naismith Trophy on February 11. 
Murray represented Canada at the 2013 FIBA Americas Under-16 Championship in Uruguay, and averaged 17 points, 6 rebounds and 2.4 steals per game in leading the team to a bronze medal.  He played for the Canadian national team at the 2015 Pan American Games, helping the team win a silver medal.  His final averages for the tournament were 16.0 points, 3.2 rebounds and 2.4 assists per game, while shooting 45.9% from the field.