Hamilton was born in Fulham, south-west London. He was born with "one and a half thumbs on one hand and half a thumb on the other". Hamilton is 5 ft 3ins tall. He was educated at a grammar school, and later read English at Downing College, Cambridge, where he was a member of the Cambridge University Light Entertainment Society (CULES).
Hamilton first came to notice while performing at the Edinburgh Festival Fringe in the 1970s.
His early radio work, mostly on BBC Radio 4 included Week Ending, The News Huddlines and The Million Pound Radio Show (with Nick Revell). He has since appeared regularly in Chelmsford 123, Have I Got News for You, The News Quiz, QI, and If I Ruled the World. Hamilton is frequently invited as a panellist on The News Quiz and as a guest panellist on I'm Sorry I Haven't a Clue.
Andy is the voice of Dr Elephant, the Dentist in the children's show Peppa Pig. He was also the original voice of Bob Fish, who is also a dentist, in the cartoon Bob and Margaret.
On 16 March 2007, he co-presented BBC Radio 4's Woman's Hour programme, alongside usual presenter Martha Kearney, as part of that day's Comic Relief fundraising activities, after defeating Richard Hammond and Kelvin Mackenzie in a poll.
Since 1995, Hamilton has written and played the lead role of Satan in the Radio 4 sitcom Old Harry's Game.
Andy toured with his UK stand-up show, "Hat of Doom" in 2008.
In 2009, Hamilton presented the BBC Four series It's Only A Theory with Reginald D. Hunter.