The President of the United States does not receive, nor has he ever received, a background check.

The only requirements to be President of the United States are laid out in the Constitution (Art. II, § 1, Clause 5). The clause reads:

No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States.

The above is the only requirement from a Constitutional perspective to become President. Other than that, it is up to the voters to decide, or more correctly, the Electoral College, which is described in Article II, Section 1, Clause 3 and amended in with the Twelfth Amendment to the United States Constitution.

While electors are pledged to a specific candidate for both President and Vice-President, there is no Constitutional requirement for them to fulfill this pledge. "Faithless Electors" do occur from time to time, such as in 1976 when an elector from Washington State cast his vote for Ronald Reagan, who had challenged Ford in the Republican Primary, rather than Ford. While some states have laws against faithless electors, no electors have ever been punished, nor is it certain these laws are even constitutional.

There, the Electoral College, and therefore the votes, indirectly, can be seen as a check on the President. If a candidate is a threat from a security perspective, they still have to actually win the election before s/he becomes President of the United States.

In addition, if issues are found with a candidate, from a security perspective or anything else, Article II, Section 4 states:

The President, Vice President and all Civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.

Whereas the impeachment process is majority vote in the House of Representatives (currently 218 votes out of 435 members) and a 2/3 vote in the Senate (67 votes out of 100 members). This is described in Article I Sections 2 and 3. This is a second check on an unqualified President, which is really what your question boils down to.

Hope this clears things up!

The base premise that I've had used in discussions in law classes when discussing security clearance and thus classification of information with respect to the president is that all classification originates with the president - that is, documents, data, and other testimony that is deemed classified is done so at the discretion or directive of the president either express or implied. Because the president in this function decides what is, and is not, classified, the president cannot in this system violate classified information protections because if he chooses to share it - by virtue of his authority as the source of classification, any information thus shared becomes declassified.

As far as requirements or qualifications for the presidency - the constitution is clear and consistent. The only requirements are the citizenship and age rules - 35 years of age, natural born citizen, resident in the country for 14 years. That's it. Any other requirements laid out would almost certainly be found unconstitutional and thrown out by the courts, and no such restrictions have been attempted as far as I'm aware.