[1][2] Born in Sydney, Dickson first played soccer in his youth before switching to Australian rules football at the age of 9.

[8] Dickson's Australian football career highlight occurred when he was chosen to play for the Swans in the 2014 NEAFL Grand Final.

In 2015, at the age of 19, he moved to Melbourne to trial with Prokick Australia, a coaching program designed to help aspiring kickers and punters cross the Pacific and crack American football.

[10] Through his Prokick Australia experience, he was picked up by the University of Texas as a punter by head coach Charlie Strong.

In late December 2017, it was announced that Dickson would forgo his senior year at Texas in favor of the 2018 NFL draft.

[20] He made his NFL debut during the season opening 27–24 road loss to the Denver Broncos, punting six times for 354 net yards.

[22] With minutes remaining at fourth-and-8 from their own three-yard line, Dickson was instructed to waste time in the end zone before giving up an intentional safety.

[33] In Week 2 of the 2020 season, against the New England Patriots on Sunday Night Football, Dickson punted four times (50-yard average) with all four of them landing inside the 20-yard line during the 35–30 victory.

[39] The "double punt" received national attention due to its relative obscurity, and is considered one of the most memorable special teams plays of all time.