Davenport was born in Sidcup in Kent, and studied art at the Northwich College of Art and Design in Cheshire before going to Goldsmiths College from where he graduated in 1988. In the same year he exhibited in the Damien Hirst-curated Freeze exhibition which brought together many of the so-called Young British Artists. Davenport's first solo show was in 1990, and in 1991 he was nominated for the Turner Prize.
Many of Davenport's works are made by pouring paint onto a tilted surface and letting gravity spread the paint over the surface. He has usually worked on medium density fibreboard rather than canvas, and most often employs household gloss paint, meaning the viewer can see their own reflection in the work. He has made a number of diptychs and triptychs as well as single works.
Davenport's use of dripped paint and gravity has been compared to similar techniques employed by Helen Frankenthaler and Morris Louis in the 1950s and 1960s.