MANCHESTER, England—Questions about British Prime Minister Boris Johnson’s past threatened to overshadow the annual gathering of his Conservative Party, as he seeks to rally support for his Brexit strategy and lay out a platform for a general election that is likely later this year.

At the party conference, Mr. Johnson is seeking to drive home the message that he will take the U.K. out of the European Union at the end of October and will boost spending on health and infrastructure projects—two promises he hopes will win voters...