Žilka was born on August 19, 1996, in Pardubice, Czech Republic. His early success at chess caught the attention of Lubomír Kaválek, a Czech-American chess player, who sponsored him.
Žilka won various international tournaments. In 2018 he won the Open Polish Championships in Poznań, and in 2019 he finished first in the GM tournament in the Czech town of České Budějovice. Later that year he achieved another major success: a shared 1st–4th place in the European Championship in Skopje.
He represented his country multiple times in international competitions such as the European Team Chess Championships and the Chess Olympiad. Also, he has been a member of the chess club Unichess, seated in his hometown of Pardubice.