Graham came through the ranks at his hometown club, Hawick, where his pace and attacking ability first stood out on the annual Kings of the Sevens circuit. He was called up to play for Hawick in the Scottish Cup Final at Murrayfield aged just 17.

An astute, attack-minded player, Graham scored two tries in eight caps for Scotland U20. These included a sensational score against Australia at the 2015 World Rugby U20 Championship, which helped Scotland to their first ever win over the Wallabies at that level. He also starred for Scotland 7s before joining Edinburgh in 2017.

Graham earned his first Scotland cap off the bench against Wales in Cardiff in the Doddie Weir Cup in November 2018. After returning from injury early in the 2018/19 campaign, he scored memorable tries in back-to-back Heineken Champions Cup wins over RC Toulon and Montpellier, which helped Edinburgh reach the quarter-finals. He finished that season with seven tries. His brace against England in the 38-38 Calcutta Cup draw at Twickenham put him firmly on the international map.

He kicked off the inaugural United Rugby Championship with two tries in a bonus-point win over Scarlets. He then went on a run of four tries in four Tests in the 2022 Autumn Nations Series, including a hat-trick against Argentina. At Rugby World Cup 2023 in France, he scored four tries in the 84-0 win over Romania.

On Christmas Day 2023, Graham committed his future to the capital with a new three-year deal. Soon afterwards, an injury in January 2024 kept him out for eight months. He returned in style that autumn, scoring four tries against Fiji on his first Scotland appearance in 13 months, then equalling the national try-scoring record against Portugal a week later.

In May 2025, his hat-trick in the 47-17 win over Ulster at Hive Stadium secured Edinburgh’s place in the URC play-offs and a return to Champions Cup rugby. That summer, he was called into the British & Irish Lions squad in Australia, becoming Lion #884 and scoring against the First Nations & Pasifika XV.

Graham won his 50th Scotland cap against Argentina in November 2025. In March 2026, he scored twice in the 50-40 Six Nations win over France at Murrayfield, becoming Scotland’s outright all-time leading try-scorer. He has since added to that record, and also earned a first call-up to the Great Britain Sevens side for the Bordeaux Sevens in June 2026.