Brinkley was born in Woodbridge, Suffolk. In 1792 he became the first director of the Dublin observatory. His main work was on stellar astronomy and he published his Elements of Astronomy in 1819. He was awarded the Copley Medal by the Royal Society in 1824.
Brinkley's observations that several stars shifted their apparent place in the sky in the course of a year were disproved at Greenwich by the Astronomer Royal John Pond.
In 1826 he was appointed Bishop of Cloyne in Cork. Brinkley died in Dublin.
Brinkley was succeeded at Dublin observatory by William Rowan Hamilton.