At the request of EW, I’m posting my expanded analysis of the Dylan Roof background check matter, along with some exposition on the whole background check procedure and underlying law. Please note, though, that the story keeps moving as more information comes into the public domain. Moreover, please note that (a) while I’m discussing the law, this is not legal advice, and (b) I have not delved into the depths of the federal regulations interpreting the statutes involved.

The Dylan Roof background check story is getting murkier and murkier.  It does, however, seem clear that he did pass the background check. [As noted below, later-developing information shows it was through the 3-business-day provision.]  Whether he should have or not is something of an open question the answer to which depends on:
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(a) the charge he was facing,
(b) how FBI and the insta-check people coded that charge v. the requirements of the federal gun control statute,
(c) whether the coding was accurate/correct,
(d) whether the charge was accurately entered into the database,
(e) whether things were done accurately and within proper procedure, and
(f) whether he was truthful on his form 4473. 

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We’ll start at (f) and work backwards. Take some time and look at the form 4473. https://www.atf.gov/file/61446/download
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This is the form a purchaser fills out when they want to buy a gun from/through a federal firearms license holder, e.g., at a store which sells guns. It is used to effectuate the federal instant background check. While it does mention states, that’s in regard to states which take care of running things through the instant background check in order to effectuate the states’ own background check requirements. It does not otherwise address state law, nor do I.
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Since we know Roof got the gun from a store, the procedure in states like the one where I live – which has no state-level gun purchase laws beyond requiring proof of residence and passing the federal background check – which I assume is similar to South Carolina’s goes something like this.
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1. The buyer goes into the gun dealer’s physical store and picks out the gun he wants to buy, reaches an agreement with the store as to the price.
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2. The store’s representative – the person behind the counter who is authorized on the store’s federal firearms license to do transactions – slides a copy of form 4473 and a pen across the counter and takes some government-issued ID (photo ID is strongly preferred if not required) from the buyer and ascertains that the buyer is who he/she says and also that the buyer is not buying for someone else. If the buyer is buying for someone else (including things called “straw purchases” but also things called “Christmas gifts”, neither of which is in issue here) that someone else gets a 4473 to fill out – they have to pass the background check.
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3. The buyer fills out the 4473, in ink. Some high-volume stores have access to an electronic form 4473, which is filled out on-line.
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4. The buyer signs the completed form 4473 and hands it across the counter to the seller. Seller checks it initially for completeness and for some of the answers to the questions: question 11 and its subparts address things which disqualify the buyer from possessing a firearm and, if you answer “yes” to any of them the instructions on the form tell the seller to stop the transaction before even calling the instant check folks – the transaction cannot proceed because the buyer cannot take possession of the gun.
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5. If the transaction appears from the form to be ok to go forward, then the seller goes to the back office and makes a phone call to the instant background check line. The seller gets a live person on the other end, gives them the FFL identifier number, his name, and a codeword unique to him and his store (this, to ensure it’s really him and not someone masquerading). Passing this test, the seller then gives the background check operator information from the form about the buyer and the gun (long gun or handgun), including the buyer’s name and SSN.
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6. The background check operator runs this information through their database and comes back with an answer.

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The background check people can give 4 responses: “Proceed”, “Cancelled”, “Denied” or “Delayed”. Those are the words they use. Only if they give “Proceed” does the transaction go through immediately. “Cancelled” and “Denied” stop the transaction permanently. “Delayed” gives the government 3 business days to decide to allow or disallow the transaction. If they do not decide within 3 business days, then the transaction can go forward.
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The point of the 3 business-day provision is to force the government to act.
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Note that, until the instant background check operator comes back with “Proceed”, neither the money nor the gun change hands. Once the answer comes back “Proceed” or, after a response of “Delayed” 3 business days pass with no response from the government, only then can the money and gun change hands.
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Why this rigamarole? This was developed through a lot of negotiations leading up to the 1994 law which implemented it. There was a concern that the government would use the information derived from the background checks to create a database of who owns guns. History showed that almost uniformly such registries were followed sooner or later by confiscation, anathema to gun owners. I and most other people who really think about it have a hard time believing there is not such a database developed already but that’s another story.
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I’ve gone through this a number of times and, because I have no arrest record and a clean (non-existent) criminal record, as well as an uncommon name, it only takes a few minutes. You can imagine, though, that someone with a common name, “John Smith” or such, might take a bit longer, as well as someone with an arrest record or criminal conviction(s).
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What seems clear by now about Roof is that the response from the background check was “Delayed”, and that the three-business-day period elapsed with no resolution, resulting in Roof being allowed to “Proceed”. The whys behind that delay are bound to be an epic exercise in finger-pointing among the various federal, state and local law enforcement agencies.
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We can, however, look at what we “know” from press reports and compare it to the rules. In Roof’s case, Questions 11 (b), (c) and (e) on the 4473 are most pertinent.
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If the charge he was under carried a sentencing exposure of 1 year or greater (my sub (a), (b) and/or (c) above), they should not have allowed the transaction to proceed, i.e., come back with “denied” during the 3-business-day period. It appears that there was some confusion on the nature of the charge Roof was facing – whether it was a “felony” or other charge with sentencing exposure greater than 1 year. That screw-up is on the government.
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If the background check folks made a mistake on the exposure under the statute he was charged with violating (my sub (d) above) or made a cross-referencing error (my sub (e) above), then it’s on the government.
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If he lied, he’s just exposed himself to serious federal time. [After SC gets done executing him, of course.] There was a Supreme Court case on this point within the last year or two.
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If he was under a charge of using pot, answering 11(e) truthfully could be problematic – he’d be within his rights to decline to answer on his 5th Amendment rights if the charge involved using pot.  Of course, declining to answer any of the questions on the 4473 is supposed to result in the seller stopping the transaction before even calling the instant background check line. OTOH, Roof would also have been within his rights to say “No, I am not an unlawful user of pot”, in accordance with having entered a “not-guilty” plea.
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You will note, if you go into the definitions and such further down on the 4473, that there is no definition given for “unlawful user of or addicted to…”  yadda yadda.
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In the same vein, it appears (thank you, EW) he may have been busted with methadone. I have not looked into the regulations -which might define “unlawful user or addicted to”, etc. – but since the only legally-recognized use for methadone I’m aware of is to ameliorate opioid addiction, though it is also widely abused (i.e., “unlawful use[r]”), a careful look at the charging documents might have shown him to be ineligible under 11(e). In other words, being charged with a drug offense, particularly methadone, might have been an automatic DQ under 11(e) regardless of how he answered, even assuming a “no” answer was not a “lie”.
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But, it appears the government missed this. They seem to have gotten screwed up on where jurisdictional lines are drawn – his arrest was near a jurisdictional line – and looked in the wrong PD’s records. They may have also possibly screwed up on the nature of the charge. And whether it was a felony or greater than one year exposure.
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So, to the extent Roof slid through, it seems to have been on the government.
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Setting all that aside, I’d like to close by asking you to consider a larger, though more subtle, issue. There exists a bureaucracy created to track allegations and convictions of criminality and decide on the relative merits of people, all within minutes or even seconds. This bureaucracy has to track every person in the US. This is all in addition to the surveillance the government is already doing. It’s of a piece with with the E-Verify system now in place for employers getting government permission to hire you – making sure you’re eligible to work in the US. Do you really like the idea of the government having that much power over your life?