I don't remember them giving him an origin in the cartoon
In the comic, Snake Eyes had surpassed Storm Shadow in the use of the sword, but Storm Shadow was extremely gifted with the Bow and Arrow.
They depicted him shooting an arrow and hitting a target in the bullseye and continuing through a wall and hitting a squirrel that was on the other side.
They used this example of Storm Shadow's disregard for life as starting to drive a rift between him and Snake Eyes.
The hard master was killed by an arrow while he was imitating Snake Eye's breathing pattern during a demonstration of the blind sword technique.
At first we're led to believe it was Storm Shadow, because he was the only one gifted enough with the bow to accomplish that.
We find out later on that it was Zartan using a technology assisted compound bow & disguised as Storm Shadow who actually killed the Hard Master.
Storm Shadow joined Cobra so that he could infiltrate their ranks to hopefully find out the identity of his Uncle's assassin.