A key figure at the heart of the burgeoning impeachment probe is former New York Mayor Rudy Giuliani, who as personal attorney to President Trump pressed Ukraine on pursuing an investigation of one of his boss’ political rivals.

A whistleblower complaint released Thursday depicts Mr. Giuliani, 75 years old, as eager to thrust himself into U.S. foreign policy. In some instances, he acted on his own, and in others his actions were in conjunction with U.S. government officials.

Ukrainians...