Kukleta started to play his club football in Czechoslovakia for Zbrojovka Brno and RH Cheb. In 1986, he was awarded the Křišťálovou kopačku award for goal of the year, for his scissors kick in Brno's second division match against Slušovice.[1] In 1988, he joined Sparta Prague, with whom he won the league three times in three seasons, as well as the Czechoslovak Cup in the 1988–89 season.[2] He was the top scorer of the 1990–91 Czechoslovak First League with 17 goals.[3]
In 1991 Kukleta moved to Spain to play for Real Betis, where he joined compatriot manager Jozef Jarabinský, as well as players Michal Bílek and Alois Grussmann.[4][5] He scored six goals in 28 appearances for the Spanish side.[5] Kukleta later returned to his homeland where he played for Brno for the rest of his professional career.[2] He amassed a total of 159 appearances and 58 goals (counting First League only) in his professional career.[2] Kukleta went on to play in the lower leagues for Děčín, Zeman Brno, Mutěnice and Velké Meziříčí.[1]