Because I opted to take Mewtwo from the video games rather than the anime or Pokemon Special (the former because I'm not much of a fan of the anime, even if the first movie was good, and the latter because I hadn't read it when I apped Mewtwo), he's admittedly more-or-less a canon OC. As a result, he's steeped in both canon and fanon information and things I've used to fill in the cracks. This post is to show what his background is and to show what differences there are between it and the non-video game source materials.
Mewtwo's story begins with Mew being discovered and captured by Doctor Fuji and Gym Leader Blaine (while the anime has Fuji as the only researcher and the manga has Blaine as the only researcher, Pokemon RBGY shows that the two were colleagues and more likely than not worked on the Mewtwo project together).
Mew is described as the Pokemon from which all others evolved from. This conflicts with later games, most prominently with the D/P/Pt and the existence of Arceus, who created the universe, Pokemon, and humans from nothing. To mediate this from being a conflicting plothole, I simply assume that if Arceus is God, then Mew would be the Pokemon equivalent of both Adam and Eve. Basically, all Pokemon besides Legendaries (because the majority seem involved with the creation of the world itself) evolved from Mew, explaining how Mew could have the DNA of 'all' Pokemon while allowing Arceus to have created the world.
After Mew was taken from the jungle it lived in to Cinnabar Island, Blaine and Fuji for whatever reason set about to create the most powerful Pokemon in the world by cloning and genetically engineering Mewtwo from Mew's DNA. Although there is no information in the games as to why the two scientists felt motivated to create a living weapon, I'd willingly assume it was funded by Team Rocket under the basis of Why Not; along with that, both the anime and the manga attribute Team Rocket as the reason he was created and, considering Team Rocket's influence and goals in the video games, it's not much of a stretch. Who else is there in the game with both the resources and the motivation to want something like Mewtwo created? On top of that, aside from the creation of Mewtwo, both Blaine and Fuji are described as kind people.
That said, this Mewtwo is completely unaware of who funded and motivated the project.
Based on the information in-game along with taking a few creative liberties, I've assumed the genetic engineering took YEARS. These experiments, as the Pokedex entries describe them, were agonizing and vicious, which left Mewtwo in a constant state of agony. It can also be assumed he wasn't the sole attempt at this experiment, because there are always failures before a success; as a result, Mewtwo had companions who ended up dying as a result of not being as strong as himself and incapable of handling the experimentation.
To borrow from the anime, I'm assuming Dr. Fuji also attempted to apply this cloning technology to his deceased daughter and failed, adding one human to the list of Mewtwo's companions who died.
Eventually, Mewtwo escaped the laboratory on Cinnabar and destroyed it. Both Fuji and Blaine survived, obviously, because the player encounters Blaine in his gym and Fuji in Lavender Town. Blaine seemed to have given up his traveling and science experiments in favor of being a Gym Leader and making simple riddles for people to solve; Fuji left Cinnabar entirely to live right next to Pokemon Tower, so perhaps he did this in hopes of compensating for the bad things he had done by creating Mewtwo? Regardless, Mewtwo left Cinnabar and eventually found the Unknown Dungeon, which contained the most powerful wild Pokemon in all of Kanto.
Here, Mewtwo split his time between meditating/sulking motionlessly to conserve his energy and fighting (annihilating) the wild Pokemon in the cave. While the Pokemon here were the strongest on the continent, he was still well beyond them in regards to power. Due to the horrific experimentation and the loss of his companions, Mewtwo became a broiling ball of rage and hate, which he kept under control so as to release it during battle to maximize his power. Due to his horrific, carnal, berserker style of fighting and the fact it was fueled by rage and pain, Mewtwo--in every single generation of games--is described as having the "most savage heart of all Pokemon".
Eventually, his training and biding-of-time was interrupted by two trainers who entered Unknown Cave: Red and Leaf. I opted to assume both entered because I'm reluctant to choose which one would be the canon player character to meet Mewtwo. Only one was the Champion, but Mewtwo didn't know or care. The two of them engaged Mewtwo in battle and managed to defeat him. The fight collapsed the cave and trapped Mewtwo--I'm basing this on RBY/GSC, rather than the newer remakes, so Unknown Dungeon was destroyed and all that was left of Mewtwo was a piece of his DNA called the Berserk Gene, which could infect other Pokemon, allowing their powers to be maximized for a short time at the expense of losing their concentration during the time.
I took Mewtwo immediately after his battle with Red and Leaf.