Bleackley was selected by the Red Deer Rebels in the 1st round of the 2012 WHL Bantam Draft from AAA Bantam club the Okotoks Oilers. After playing a further half season of midget hockey with the UFA Bisons of Strathmore in the Alberta Midget Hockey League he made his WHL debut in the 2011–12 season. In only his second full season with the Red Deer Rebels he was named team captain as a 17-year-old for the 2013–14 season. Bleackley whilst showing positional versatility broke out offensively with the Rebels, producing a career high 29 goals and 68 points in 71 games. His outstanding play was further rewarded when he was chosen to skate in the 2014 CHL/NHL Top Prospects Game. Bleackley draft stock rose and he finished the season rated as a top prospect who was viewed as a possible first round selection heading into the 2014 NHL Entry Draft before he was selected by the Colorado Avalanche. In Bleackley's first NHL training camp, the Avalanche were critical of his conditioning and promptly returned him to the WHL without an entry-level contract. In the 2014–15 season, Bleackley continued his scoring pace from previous seasons with 27 goals and 49 points in 51 games. On February 6, Bleackley suffered a strained groin, effectively ruling him out until the post-season in which Red Deer suffered a first-round exit. During the off-season in the lead up to the 2015–16 season, Bleackley returned to the Avalanche rookie and training camp with ambition to earn an entry-level contract. With an improved physical performance, Bleackley was later returned for his final season of junior after competing in a pre-season game with the Avalanche. Prior to the commencement of the WHL season, Bleackley was relinquished of the captaincy by the Rebels, with coach Brent Sutter explaining the intent to focus on reaching his potential. With a guaranteed berth at the 2016 Memorial Cup, Bleackley struggled in the early stages of the year offensively. Bleackley picked up his offensive production and scored 41 points in 47 games before suffering a broken kneecap against the Everett Silvertips on January 23, 2016. With the Avalanche intending not to offer Bleackley a contract, at the trade deadline on February 29, 2016, his rights were included in a trade alongside Alex Tanguay and fellow prospect Kyle Wood to the Arizona Coyotes in exchange for Mikkel Boedker. Citing his injury history, the Coyotes also declined to offer Bleackley a contract and instead chose to receive a compensatory second-round draft pick in the 2016 NHL Entry Draft; Bleackley will be required to re-enter the same draft. After his re-selection in the 2016 NHL Entry Draft by the St. Louis Blues, on July 2, 2016, Bleackley was finally signed to a three-year, entry-level contract. At the conclusion of his entry-level contract, Bleackley was not tendered a qualifying offer by the Blues, releasing him to become a free agent on June 25, 2019. Unable to attract NHL interest, Bleackley was signed to a one-year ECHL contract with the Idaho Steelheads on September 3, 2019. He was invited to attend the Steelheads AHL affiliate, the Texas Stars training camp prior to the 2019–20 season. On October 5, 2019 after a successful camp he was signed to a professional try-out contract with Texas to remain on the roster. Bleackley opened the season with the Stars, scoring 2 goals in a 5-3 defeat to the Manitoba Moose on October 5, 2019. Showing initial success with the Stars, Bleackley remained with the club through 10 games, registering 5 points, before he was returned to the Idaho Steelheads. Bleackley compiled 13 points through 18 games with the Steelheads before he was dealt in a three-way trade to the Maine Mariners on January 14, 2020.
