In general, the point of electing someone is that the voters are supposed to decide their qualifications. When there are restrictions on who can hold an office, the restrictions tend to be very simple -- are you a citizen, are you a resident of the area, have you been convicted of a felony, etc. Some positions might require certain professional qualifications (e.g. requiring a prosecutor to be a licensed lawyer), but again, those are fairly simple yes/no questions and are requirements you'd probably have met long before running for office.
At the federal level, the only background requirements are those set out in the Constitution. Representatives must be at least 25, Senators at least 30, Presidents at least 35; Representatives must have been a citizen for 7 years, Senators 9, Presidents their whole life; Representatives and Senators must be residents of the state they represent; someone who swore to uphold the Constitution and then took part in a rebellion against it can't hold any office at any level of government; and someone who's been impeached and removed from federal office can be barred by the Senate from ever holding any federal office again (I don't think this applies to being in Congress but does apply to being President, but am unsure). Per Powell v. McCormack, the constitutional requirements for Congress are the only requirements for Congress; I believe the same logic would be extended to the Presidency if it ever came up.
The kind of background check a private company does, where they subjectively decide based on your full background whether or not you're suitable, is not used for elective office. The closest you'd get is if you need a professional license (like a prosecutor needing to be a lawyer), where the license might have some subjective "good moral character" requirement, the only requirement for the office is that the licensing agency has licensed you.
That's not to say there's no subjective background component. Both the opposition and the press are able to dig into your background and report whatever they find. That lets the voters decide whether or not your background is disqualifying. The whole point of putting an office up to a vote is that you trust the voters to decide who is suited to hold that office. For instance, at the time I'm writing this US Senate candidate Roy Moore has been accused of sexual misconduct back in the 1970s. This is the sort of thing that will basically never show up on a normal background check; however, the media attention on the Senate race meant people were probing deeply into his history, resulting in accusers coming forwards. Now that the allegations are public, the voters of Alabama get to decide whether or not they want to elect someone who's been accused of these things.
