Mr. Mulvaney was elected in the Tea Party wave of 2010, when he defeated John Spratt, a veteran Democratic congressman who had been chairman of the House Budget Committee. He won by branding his opponent a big-spending liberal, unconcerned about fiscal prudence.

Once on Capitol Hill, Mr. Mulvaney joined a conservative bloc that pressed for slashing federal spending more deeply than House Republican leaders preferred, and became a prominent face of the anti-Washington movement on Capitol Hill. He was one of several dozen House Republicans who refused to back the deal to raise the statutory debt limit.

Mr. Mulvaney has repeatedly opposed some of his own party’s budget proposals, and quickly established himself as one of the most outspoken members of that 2010 class of Republicans. By 2013, at the outset of his second term, he declined to support Mr. Boehner’s re-election as speaker, abstaining from the vote in protest.

Strongly anti-establishment, Mr. Mulvaney, who has a degree in international economics from Georgetown and a law degree from the University of North Carolina at Chapel Hill, chafed as much at the Republican leadership in the House as he did at Mr. Obama’s direction from the White House.