Born in Tokyo, Graham attended day and boarding schools in Australia and the UK, including Eton College, as his family moved around for his father's work.

[6] His television appearances include The Stand-Up Sketch Show (ITV2), Live at the Apollo, Mock the Week, Live from the BBC, Russell Howard's Stand Up Central, As Yet Untitled, Virtually Famous, QI, Sweat the Small Stuff and The Dog Ate My Homework, and he is a regular on radio show Fighting Talk.

[7][8][9][10] In October 2019 and December 2021, he competed on Richard Osman's House of Games[11] and appeared on Paul Merton's team on Have I Got News For You on 8 November 2019.

[18] In 2019 Graham was included in the i-news "50 of the best jokes of the Edinburgh Fringe" with his line: "Do I enjoy randomly appointing people to judicial positions?

In late 2021, Graham co-presented the Dave travelogue British as Folk alongside fellow comedians Darren Harriott and Fern Brady.