Kirkeskov was born in Aarhus, Denmark, and started playing football for VRI before joining the youth ranks of the city's main team, AGF in 2005.[2] He was promoted to their first team in the spring of 2009.[3] He made his league debut for AGF in the second tier of Danish football in a 3–0 away victory against FC Fredericia on 7 November 2010, replacing Adam Eckersley at half-time.[4] After the winter break, Kirkeskov made more appearances and the club reached promotion back to the Danish Superliga towards the end of the season.[5] He made his debut in the top tier on 15 August 2011 in a 1–1 away draw against Silkeborg IF, coming off the bench in the 80th minute for Stephan Petersen.[6] In his first season in the Superliga, Kirkeskov made 16 games.[7]
In December 2020 it was announced Kirkeskov would move to 2. Bundesliga club Holstein Kiel for the second half of the 2020–21 season on a contract until the summer of 2023.[11]
Zagłębie Lubin
On 31 July 2023, Kirkeskov returned to Poland as a free agent, joining Ekstraklasa side Zagłębie Lubin on a two-year deal.[12] He made nine appearances before terminating his contract by mutual consent and forfeiting the remainder of his salary on 18 January 2024, citing personal reasons.[13]
Second stint at Holstein Kiel
Kirkeskov returned to former club Holstein Kiel in January 2024, having agreed a contract until the end of June.[14]