Graham was born in Blackburn. He attended and graduated from the University of Brighton where his stage career started. He wrote, produced and appeared in the BBC Two television series Dr. Terrible's House of Horrible (2001) and seven series of the BBC Three sitcom Ideal (2005-present). Duff has also contributed additional material to puppet comedy Combat Sheep.
Duff created, wrote and appeared in the BBC Radio 4 science fiction comedy Nebulous (2005, 2006 and 2008). Duff appeared in Exile — an audio drama based on Doctor Who , as well as writing another — Faith Stealer.
In 2010 he made an appearance in Harry Potter and the Deathly Hallows as a Death Eater. He had a notable speaking, yet uncredited, role.
He also edits the Radio 4 comedy Count Arthur Strong.