This is a very common misconception regarding background checks.
The way that people get a background check is that there is an index of prohibited persons, based on a variety of factors. If you ever end up on the prohibited persons list for these factors, you are on this list permanently, barring a judiciary hearing which appeals the decision (expensive and difficult).
The following people are prohibited persons;
A person who has been convicted in any court of a crime punishable by imprisonment for a term exceeding one year or any state offense classified by the state as a misdemeanor and is punishable by a term of imprisonment of more than two years.
Persons who are fugitives of justice—for example, the subject of an active felony or misdemeanor warrant.
An unlawful user and/or an addict of any controlled substance; for example, a person convicted for the use or possession of a controlled substance within the past year; or a person with multiple arrests for the use or possession of a controlled substance within the past five years with the most recent arrest occurring within the past year; or a person found through a drug test to use a controlled substance unlawfully, provided the test was administered within the past year.
A person adjudicated mental defective or involuntarily committed to a mental institution or incompetent to handle own affairs, including dispositions to criminal charges of found not guilty by reason of insanity or found incompetent to stand trial.
A person who, being an alien, is illegally or unlawfully in the United States.
A person who, being an alien except as provided in subsection (y) (2), has been admitted to the United States under a non-immigrant visa.
A person dishonorably discharged from the United States Armed Forces.
A person who has renounced his/her United States citizenship.
The subject of a protective order issued after a hearing in which the respondent had notice that restrains them from harassing, stalking, or threatening an intimate partner or child of such partner. This does not include ex parte orders.
A person convicted in any court of a misdemeanor crime which includes the use or attempted use of physical force or threatened use of a deadly weapon and the defendant was the spouse, former spouse, parent, guardian of the victim, by a person with whom the victim shares a child in common, by a person who is cohabiting with or has cohabited in the past with the victim as a spouse, parent, guardian or similar situation to a spouse, parent or guardian of the victim.
A person who is under indictment or information for a crime punishable by imprisonment for a term exceeding one year.
Due to advancements in technology, background checks are conducted mostly electronically today; essentially a FFL (Federal Firearms License) holder (a person or store who has registered with the ATF to sell firearms as a business) inputs your personal information, including your social security number,m name, address, etc, and uses this to look up and see if you have been flagged as a prohibited person.
If you are prohibited, you are not allowed to purchase the firearm. If they sell it to you anyway, they are guilty of a Felony that has a pretty serious jail term. There is no exception to the above; this is a Federal Law which supersedes all local and state laws on the matter.
As for the question of where people get firearms from (when robbing stores, and the like); the BJS.gov and many criminologists suggest a significant amount of firearms which are used in crimes or obtained by prohibited persons are stolen, either from family members, or from unaffiliated individuals. Any person with the above charges can only obtain a firearm illegally; even in cases of a private purchase, they are still committing a felon by buying the gun (and the person selling the gun, if they know that the person they’re selling to is a felon, is also guilty of a Felony).
As for gun violence itself; I would argue that there are root causes of violent activity, which our government and culture are either refusing to address, or not addressing well enough. It’s been shown that abusive and neglectful households result in children who grow up to be violent adults; it’s also been shown that on average a violent criminal has a lower IQ and lower educational achievement and aptitude, it’s been suggested by many that these two factors have a major contribution to not only violent behavior, but addictive behavior.
I would argue that until we, as a people, address these root causes, we will always have violence and violent crime as an epidemic, whether guns are allowed for civilian ownership or not.