Donald Trump has assembled a cabinet and senior staff with divergent views on such issues as the deficit, trade, climate change, and Russia, posing a challenge for his administration as he tries to mold his own sweeping campaign themes into specific policies for governing.

The president-elect’s pick for budget director, Rep. Mick Mulvaney (R., S.C.), has opposed raising the debt ceiling, but Mr. Trump has proposed steep tax cuts and large increases in defense and infrastructure spending that many economists believe will cause the deficit to grow.

Several of his cabinet selections backed a trade deal negotiated by President Barack Obama with Asian nations, which Mr. Trump opposed and has vowed to abandon.

His choice for the State Department, Rex Tillerson, has said he believes science proves climate change is being caused in part by human behavior, something Mr. Trump in the past has called a “hoax.”

Former White House officials said incoming administrations often have to grapple with contrarian viewpoints. In 2008, Mr. Obama selected two of his rivals for the Democratic nomination—Hillary Clinton as secretary of state and Tom Vilsack as agriculture secretary—despite differences displayed on the campaign trail. Mr. Obama already had tapped another rival, Joe Biden, as his running mate.