WASHINGTON—WikiLeaks, the transparency activist group that publishes secret documents, on Wednesday released records it said were obtained from Central Intelligence Agency Director John Brennan’s private email account, including a portion of his draft government security clearance background form from 2008.

An anonymous hacker over the weekend alleged that he had broken into Mr. Brennan’s AOL account and stolen emails, contacts and other information. This person released more than 2,000 email contacts he said he found on Mr....