I've been theory-crafting on my own and, through reading the lore on different champions, I have a few ideas of where the Void originated, who caused the problem with it, and how it was dealt with in the past. Please note these are all my personal speculations and nothing more. I don't even have the timelines of where the champions were released, not that that would make much of a difference, so I apologize if the timelines are off.
How it began:
In Shurima, a mage seeking power risked his life to attain a new body and more power than he could image. This mage's name was Xerath. He succeeded, but was imprisoned for his recklessness. Sometime during the ritual, however, he tore open a dimensional rift into another world: The Void. After he was imprisoned and entombed, possibly years later, the Voidborn attacked, beginning in Shurima where the rift was largest.
How it was dealt with:
The mages of Shurima, powerful and beyond that of those in modern Runeterra, fought and defeated the creatures from the Void, but it was not easily won. Many died, others converted and began worshiping the monsters, and the Voidborn were able to sow fear and devastation into the world.
After they were defeated and sent back to the Void, the followers managed to escape and founded a city to freely worship their gods. They were found out, killed in some kind of purge, and much of the history and records of the Void were erased from history.
Lesser creatures of the Void may have used these followers to manifest themselves into the world once more, but they were unable to keep their strengths. This would explain one reason why the creatures of the Void were wiped from history records: to keep others from mimicking the now dead followers.
Where the Void originated:
Shurima, of course, then Icathia. In modern Runeterra, the ruins of Icathia remain, providing those such as Malzahar, one gifted in prophecy, to recount what happened or be effected by the past. He could very well be a descendant of one of the Void followers that was missed, thus granting him a small link to the other dimension.
For Cho'Gath, my guess is that he was able to find the rift that had been created long ago and, through brute force alone, tore his way through. This did, however, leave him weakened and unable to resist the League when they found him.
Malzahar followed his visions to Icathia, where he became a mental anchor, one connected to the past and gifted in powerful magic, for a much more powerful Voidborn.
Kog'Maw, perhaps the child of this second Voidborn, manifested himself into the world. Manifested, in my opinion, means he left behind parts of himself, such as physical strength, mental capacity, and the much of his memory. By what we know of the little guy, he was a glutton in the Void, constantly eating.
Finally, Kha'Zix is a special case. I believe he may have been a parasite on one of the nomads of Shurima. His lore says he was fragile and weak, possibly either from his trip through the Void or, what I believe, he literally infiltrated Runeterra as a fetus. When he grew of age, or at least old enough to begin his mission, he killed his host, devoured it, gained its strength, then proceeded to kill any witnesses.
One last thing:
Kassadin was, like Ezreal and his lore says, an explorer of sorts. When he found Icathia, he did, unknowingly, stumble upon a secret the Void had turned to protecting with a passion, so much so they were willing to send one of their most powerful warriors and a trusted child to protect. Thanks to the League and the trials of leaving the Void, he was able to find out of the horrors amassing and waiting.
The Voidborn were not happy when a trespasser appeared on their most guarded secret. They tried to kill the explorer, but failed due to a stone left behind to fight off the Void. This stone is now with Kassadin always, and is even his passive ability in game.
Xerath unknowingly revealed Runeterra to the Void. There was a war. Icathia was built to serve the Void's inhabitants, but fell and was nearly wiped out. Due to the mages of Shurima fearing the return of Icathia, they destroyed much of their knowledge, magic included, to ensure no other followers would emerge. The Voidborn have found the rift and are picking at it, slowly sending warriors, assassins, and scouts through it, even manipulating those with powerful magic. Finally, when the Void does break through, I think Xerath and Kassadin are going to be the only ones who can fight them effectively. Xerath being the muscle and Kassadin the assassin.