Jílek began studying piano and composition as a pupil of Jaroslav Kvapil, and later studied conducting under Antonín Balatka and Zdeněk Chalabala at the Brno Conservatory. In 1937, Jílek completed his education at the Prague Conservatory, in the master class of Vítězslav Novák. From 1938 to 1949 he conducted the opera in Ostrava. In 1952, he became the principal conductor of Janáček´s opera in Brno, a position he held for 25 years. During his career Jílek frequently conducted the orchestra of the National Theatre in Prague, the Czech Philharmonic, as well as orchestras abroad. In 1978, he became the conductor of the Brno Philharmonic Orchestra.
He conducted the complete operas of Bedřich Smetana and Leoš Janáček, and also focused on Russian and Italian operatic repertoire. The recordings of his interpretations of Janáček's, Novák's and Martinů's work are available on the Czech label Supraphon. He was awarded the Orphée d'Or de l'Academie National du Disque Lyrique (Prix Arturo Toscanini-Paul Vergnes) for the recording of Janáček's opera Jenůfa in 1980.
- Malá encyklopedie hudby, p. 303
- Pozdní divoch, p. 216 (only in Czech)
- Jaroslav Smolka: Malá encyklopedie hudby. Prague: Editio Supraphon, 1983.
- Jiří Ort: Pozdní divoch. Láska a život Leoše Janáčka v operách a dopisech. Prague: Mladá fronta, 2005. ISBN 80-204-1256-5
- Fibich: The Bride of Messina CD. 11 1492-2 612 (Supraphon)
- Janáček: The Excursions of Mr. Brouček CD. 11 2153-20612 (Supraphon)
- Janáček: Destiny CD. SU 0045-2 611 (Supraphon)
- Janáček: Jenůfa CD. SU 3869-2 612 (Supraphon)
- Janáček: Orchestral Works I.-III. CD. SU 3886-3888-2 031 (Supraphon)