[3][4] Burke was ordained to the priesthood for the Archdiocese of Chicago by Cardinal Costantino Patrizi Naro on May 22, 1875.

[3] Burke received his episcopal consecration on October 28, 1887, from Archbishop Patrick Feehan, with Bishops William McCloskey and Henry Cosgrove serving as co-consecrators, at Holy Name Cathedral in Chicago.

[4] In addition, he faced attacks by the American Protective Association, an anti-Catholic and anti-Irish hate group.

Burke travelled to Rome to petition the Vatican to attach the diocese to a more established one, citing the dire conditions in Wyoming.

Pope Leo XIII appointed Burke as bishop of the Diocese of Saint Joseph on June 19, 1893.