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.

[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.

[11] 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.