The Punisher is a self-styled vigilante that is wanted by the FBI and even the CIA for the disappearance of various crime figures. He also uses lethal force on his victims, shooting first and never asking questions. Despite this, J Jonah Jameson still thought of portraying him as a hero because he's after Spider-Man a masked vigilante. This line of thinking is consistent with Jameson's thoughts on masked vigilantes seen in "Sting of the Scorpion".
The Punisher's motivation to hunt Spider-Man is driven by the idea to redeem himself in the eyes of the law. This is inspired by "The Amazing Spider-Man" (Vol. 1) #129 when The Jackal hired the Punisher to kill Spider-Man.
After Spider-Man feels responsible for the creation of Morbius' living vampire form and the many plasma drained victims, a flashback the power and responsibility part of Spider-Man's origin first seen in "Menace of Mysterio" is shown. This is the main motivation that drives Peter Parker to fight crime as Spider-Man.
This episode is loosely based on "The Amazing Spider-Man" (Vol.1) #101, the first battle between Spider-Man and Morbius. One particular scene taken directly from the comics was Peter getting irritated by a call from Mary Jane (it was from Gwen Stacy in the comic) after he just gained his six arm look.
This version of The Punisher is based on wears a blue uniform instead of a black one like in the comics. However, he still had the contemporary additions of a bandana and a trench coat.