Alexander was born in Rothbury, Northumberland. His father was a GP and farmer and his mother a magistrate and formidable campaigner for conservation. He has an older brother and sister. Met future comedy partner, and close friend, Ben Miller while studying at Cambridge in 1992. He and Ben toured the club circuit, and appeared on Saturday Night Live. In 1996 he was nominated for the Perrier Award with Ben, and in 1997 they filmed their the first of 4 series of Armstrong and Miller. In 2004, Alexander married Hannah, an events organiser for Harvey Nichols. They have four sons.