So, I went and looked at this and it is fairly confusing. The main part I can't figure out in reading the articles about it is how did he end up getting charged? I can't seem to find how that happened. 

Here's what I found in the articles about it:

So, many years ago he buys a gun, a .38 pistol, from a gun dealer, fills out the paperwork that you have to legally. So, it is a legal gun sale. His brother's wife finds the gun within 11 days of him purchasing it and takes it and throws it away in a dumpster next to a supermarket, because she fears he will use it to kill himself. 

OK, amazing that the gun was found in the supermarket dumpster, but it was, who dumpster dives looking for guns? Someone finds it in the dumpster though, and through the serial number on the gun it is traced back to Hunter, because it was a legal sale and the information was recorded. 

Years after all this happened, Hunter writes a book, a memoir, and in it he discusses his drug use. Apparently, I've not read the book, someone (I assume a Republican operative) reads his book and lines up the years he was using drugs with the year he purchased the gun (not sure if he mentions buying the gun in the book or not, I think one report says he did not but I have no idea). 

How does whomever is reading the book even know about the gun purchase unless he mentions it in the book? I mean, if it is not in the book you'd never know that. I doubt there was anything written about the gun purchase or the gun being found in a dumpster in old newspapers...but I guess, maybe there was?

Now, due to what he wrote in his book, someone (WHO?) digs up that Huner bought a gun during the time he claims to have been a drug user according to his memoir. Someone then gets the application that Hunter filled out to buy the gun and on it asks if he is using any drugs, the box is checked NO.

Alright, so then someone has to bring this to the attention of the authorities, WHO? Then the authorities, police, whomever, has to look at the gun application and whatever is written in Hunter's memoir/book. 

This seems really thin evidence to charge him because people can write anything in a book and you can't take it to be true, it is a "recollection" of a person that could be true, could be a lie, or could just be misremembered. Just ask Donald Trump who lies in his own books and calls every book about him a bunch of lies.

So, I would think no prosecutor would bring a case about a gun purchase based on something written in a book years after the purchase and the person no longer having the gun...plus he had it for just 11 days.

My question is, how did this case get brought? Who thought it was a good idea to charge him with this? From all I have seen, the only evidence they have is the memoir he wrote. I can't find anything about him getting charged with anything back when he had the gun and the case seems to hinge on the memoir he wrote. 

Does anybody know how he ended up getting charged?