Although there have been several different versions of the TMNT, through various continuities, but the basic origin remains the same. Their animosity for the Shredder is also shared across all continuities.

As the story goes, Splinter (pre-mutation) was owned by a Kung Fu master. During this time, he followed his owner in during practices and learned all about the martial arts. But, sadly, for one reason or another, Splinter's master was killed by the Shredder. Splinter broke free of his cage, made it into the sewers, ended up with some turtles, and whole shebang got covered in some glowing ooze.

The mutagenic properties of the ooze caused all the animals to grow to humans sizes and to gain human-levels of intelligence (well, Splinter already had that, what with learning martial arts, but still). Splinter taught the turtles Kung Fu, and then eventually sent the turtles to confront the Shredder.

Usually, the Shredder died.
 
Webmetz
2016-08-07 15:28:08 
And occasionally, Shredder was an Alien Octopus Thing (Utrom) named Ch'rell in a exo-suit. In other realities, an Utrom named Krang gave Shredder the Ooze which started the whole shebang. (I still like to think that Shredder was still Ch'rell)
    

When evil spreads across the land, and darkness rises and the monsters roam. When the creatures of the night make beautiful music, and the things that go bump in the night go bump with greater enthusiasm. When the world is in peril and is in need of a hero...

These guys are, sadly, the best the world can hope for. These are the adventures of the heroes of CVRPG. They mean well, they try hard, and occasionally they do the impossible...

They actually do something heroic.