According to what I've read Brainiac 1 (Superman's longtime nemesis) is father to Brainiac 2, who in turn is father to Brainiac 3...But from then on I'm unsure as to how the family tree progresses.  Brainiac 5 of the Legion of Super Heroes is supposed to be the child of Brainiac 4, but I'm a bit confused because according to comicvine the name Brainiac 4 refers to both his mother AND his father, and since Brainiac 5 is from the future I'm unsure how distantly related Brainiac 4 is to Brainiacs 2 & 3. 
 
Oh, and apparently Brainiac 8/Indigo comes from even further in the future than Brainiac 5, and there had to have been a Brainiac 6-7 as well at somepoint (so no idea how they're connected to the rest of the Brainy family tree).
 
Can anyone clear this up for me?

There is indeed Brainiac, Vril Dox Jr (that is what he is almost always known for, but he is also called Brainiac 2 only briefly), Lyrl Dox, who demands to be called Brainiac 3, and then there's Brainiac 5 in the future, and there are mentions of a Brainiac 6, who programmed Indigo, and there is also a Brainiac 13 from the 64th century. 
 
The thing is, though, Brainiac 5, and presumably all of the following Brainiacs, do not come from Lyrl Dox/Brainiac 3. In the first volume of the most recent R.E.B.E.L.S. series, a hologram of Brainiac 5 tells Vril that if he dies, the Dox line dies with him. This is after the birth of Lyrl, so it maybe means that Lyrl will die soon, or he is just not the ancestor to all the other Brainiacs, and Dox will have another child.

After Brainiac 3, the Brainiac name is forgotten for a thousand years until B5's father reclaims it in an attempt to clear his family's name, becoming Brainiac 4. He gives his son the same purpose, and he takes the name Brainiac 5.

Pre Crisis there were only 5 Brainiacs. as stated in Secrets of the Legion of Super-heroes. Brainiac adopted Vril Dox, had a son called Pran Dox who had a son called Kajz Dox who was the father of Querl Dox. This presumes Coluans lived a very long time. 
 
In reboot continuity Querl's mother was Brainiac 4.

Most people don't seem to know this, but Brainiac 5's real name is Querl Dox and is actually Brainiac's biological son whom which he experimented on and cyborized to at least save one of his sons from his dying world and experimenting on his own son against his wife's wishes was going to far. And the reason he collects knowledge is to figure out all of time and space to fix the cause of his universe's destruction. Querl Dox managed to exile himself right before his world's destruction, just as his father predicted, hoping that his father could fix everything. But by the time he escaped, it turned out that his father spend his millenniums as the destroyer of worlds with a bad reputation on the family legacy. He would then spend his life fearing what his father made him into and that he would end up becoming him and with all hope of saving his world lost for good, all while being feared by everyone else as if he was just like his father. This is why Brainiac 5 joined the League of Heroes in the future, to prove that he was nothing like his father and rejecting his father until his father's last sacrifice in repentance for his crimes, because he knew that he at least got to save the people he cared about the most. His sons, Querl Dox & his older brother Vril Dox jr, Vril Dox's sons, Lyrl Dox and Pran Dox (nephews), and his wife, Brainia Starr.

The nick names 1,2,3,4,5, etc... are given according to the order of which characters where known at the time. So because Brainia Starr was never known until after Brainiac 3's defeat, she was later known as the forth Brainiac themes villain, or Brainiac4. But then their is the Male Brainiac 4, which I am not familiar with.

DISCLAIMER: I am not sure if this is currently ret-coned from continuity.

Despite popular belief and movie/show adaptations, not all Brainiacs are artificially created. The Brainiac family is an actual family; however, this has been changed in several adaptations so that they would all be the same original character as a series of upgraded bodies, the post popular examples being the versions of Brainiac where he is basically just a Terminator type of robot who was created as apposed to born.