Before meeting the turtles, back when Yoshi lived in Japan, there was a ninja clan known as The Foot, which was led by Yoshi. One of his students, Oroku Saki, constantly challenged him in an effort to usurp his leadership, finally causing him to be banished from the clan. Yoshi fled to the United States in poverty. He was forced to live in the sewers, where he befriended the rats. One day, a small child was returning from a pet store, carrying four baby turtles in a fish bowl, when he tripped and fell, causing the bowl to crash on the street and the turtles fell into a drain. Yoshi had found four new friends. One day, Yoshi discovers the turtle are covered in "goo", which he later realizes was actually some kind of mutagen that caused whoever touched it to take the form of the animal it most recently came in contact with. The turtles became humanized, with their recent contact with Yoshi (with his attempt to clean them up), and Yoshi becomes a mutant rat with his recent contact with rats.
Since then, Hamato Yoshi raised the turtles by himself, and gave them the names of his favorite Renaissance artists: Leonardo, Donatello, Raphael and Michaelangelo, who were to be distinguished by personality and headband color. He also taught them the art of ninjutsu in order to protect themselves better, since people above ground would not accept them.
Hamato Yoshi was given the nickname "Splinter" by his students, due to his proficiency at breaking wooden boards. On one occasion Donatello was able to use the remnants of the ooze that transformed them to restore Yoshi's human self, but the process was only temporary and soon wore off; however, Splinter, having seen how humans could treat each other at times, concluded that he preferred his new state, particularly due to the relationship he had with his students.
Splinter does not hold a father/son relationship with the turtles, as the Mirage Studios, Movie, and 2k3 stories suggested, but more of a teacher/student bond. . In fact in Turtles Forever the old toon turtles are surprised when the 4kids turtles call Splinter father. However, in an episode titled "The Old Switcheroo", Splinter showed a glimpse of fatherly concern for Leonardo when he was slightly injured by one of Donatello's contraptions. Splinter also mentions in the series (episode: "Snakes Alive!") that Leonardo is his most gifted student. Splinter even gets the chance to be human once more in an episode Splinter No More, but realizes he much more prefers being with the turtles. Master Splinter is always there for the Turtles to give a wise/grasping quote or speech to try and ease the turtles' troubles. He'll also show a little ninja action just in the nick of time when it seems the turtles are in for impending doom that is inescapable.
This Splinter had a brief role in Turtles Forever where he meets the Turtles from the 2003 TV series and was seen giving 2003 Leo and Raph some rice to nourish their bodies and souls. '03 Leonardo commented during their time with this alternate Splinter that it feels right to be there with Splinter regardless of the differences between the two worlds. Splinter's assurances that he feels the same, coupled with his comment that Leonardo and his brothers will always be welcome there, helps Leonardo see the common similarities between the teams, regardless of their different styles and methods. He never mentioned to them that he was Hamato Yoshi at all and after that he's never seen again.