Kirker was born in County Tyrone, Ireland. He moved with his father to Lancaster, Pennsylvania in 1779. He moved with his new wife to Kentucky in 1790 and then Liberty Township, Adams County, Ohio three years later. He served in the first Ohio House of Representatives in 1803 and then in the Ohio State Senate from 1804-1815. While serving as Speaker of the Senate, Kirker became Governor upon the resignation of Edward Tiffin to take a seat in the U.S. Senate. Kirker's term was extended through the 1807-1808 meeting of the Assembly due to the disqualification of Return J. Meigs, Jr. who had won the 1807 election to the governorship but had been disqualified by the Assembly as he had not met the residency requirements. Kirker ran for re-election in 1808, but lost badly to Samuel H. Huntington. Kirker later returned to the Assembly, serving in the House of Representatives from 1816-1817 and in the State Senate from 1821-1825.
Kirker left Kentucky (whether for economic reasons or because of opposition to slavery is uncertain) and moved to Manchester, Ohio, around 1793.
During Kirker's first year as acting governor the settlers in the western section of the state were alarmed by the threat of an Indian uprising.
Kirker held a position similar to Worthington's and divided the vote of those who opposed a strong court; Kirker and Worthington received 3,397 and 5,601 votes respectively to 7,293 for Huntington.