Some TMNT stuff was always meant for grownups.
In the original comics, Splinter was the pet rat of a ninja named Hamato Yoshi in Japan. Intelligent for his species, in his cage Splinter was able to learn his master's art by mimicking his movements while he practiced. Yoshi became embroiled in a dispute with a fellow ninja by the name of Oroku Nagi. When Yoshi had to go to America into exile, years later he was eventually murdered at the hands of Nagi's brother Oroku Saki, who sought revenge for the death of his brother Nagi.
Without a home, Splinter was forced to run away and live in the New York City sewers. Due to a traffic accident, four baby turtles and a canister of radioactive ooze were sent down into the sewer, where they were exposed to the strange substance that mutated both Splinter and the turtles. Within hours Splinter and the turtles were intelligent, humanoid mutants. Splinter named the four turtles: Leonardo, Donatello, Raphael, and Michaelangelo (after a book on Renaissance artists that he found in a storm drain) and in the following years trained them in the arts of Ninjutsu so that they should take revenge for his dead master, while still raising them as his own sons. Although Splinter at the beginning was very careful to keep his existence and his foster sons secret from the rest of the world rates, the five little turtles made friendly contacts with the upper world, in particular with April O'Neil and Casey Jones.
Splinter was usually more in the background instead of at the forefront. He stood aside the Turtles always with advice, but when it came time for a direct fight, he showed despite his age a high skill as a ninja master. In volume 4 issue #10, it would have appeared that this version of Splinter had died of a heart attack. For all his friends, it was a big shock when Splinter unexpectedly died; much later it turned out that he had found a new home in the Battle Nexus. How he got there however, remains unclear. However, as revealed in Teenage Mutant Ninja Turtles issue 32 (volume 4), the Splinter that died was not the Splinter that raised and trained the Turtles.
In V4 #26, Splinter's full formal name is revealed to be Hamato Splinter, taking the same Japanese surname as his master Hamato Yoshi.