Zelinka was born on 23 February 1802 in Vyškov, Moravia.
From 1821 to 1825, he studied law at the University of Vienna and received his PhD in 1829.
As mayor, he participated in planning the First Vienna Mountain Spring Pipeline, the Wiener Donauregulierung (a flood control project) and the Zentralfriedhof (Central Cemetery).
His entire annual salary of 12,000 florins was donated to charity.
In 1869, a street in the Innere Stadt was named the Zelinkagasse in his honor.