On a snowy Christmas night, Tucker and Esther Cobblepot throw their deformed infant child Oswald into Gotham River, fearing he would become a menace to society after attacking their pet cat. His crib floats to an abandoned zoo and is found by a flock of penguins who raise him as one of their own.
33 years later, during the lighting of Gotham City's Christmas tree, a villainous gang of carnival performers stage a riot. While Batman subdues the criminals, corrupt businessman Max Shreck falls through a trapdoor and is brought to the underground lair of the grotesque crime boss known as The Penguin (he is Oswald as an adult). A former sideshow freak, The Penguin explains his desire to become a respected citizen of Gotham and blackmails Shreck into helping him.
Meanwhile, Shreck's secretary, Selina Kyle, inadvertently discovers her boss's plan to illegally monopolize Gotham's supply of electricity. To protect his secrets, Shreck pushes her out of his office window. Falling through several canopies, Selina miraculously survives but lies unconscious in an alley. A group of cats swarm around her, and while attempting to eat her, they revive her, gifting her with eight more lives. Traumatized, Selina develops dissociative identity disorder and, after having a mental breakdown and trashing her apartment, she fashions a black vinyl costume and whip, becoming the formidable Catwoman.
Shreck arranges for one of Penguin's men to kidnap the Mayor's infant son, allowing the Penguin to "rescue" him. As a reward, the Penguin is given access to the Gotham City Archives, where he learns his real name, and that he is the last surviving member of his family. Meanwhile, the Mayor, persuaded by Bruce Wayne, refuses to give Shreck a construction permit for his power plant. The Penguin orders his gang to attack downtown Gotham, ruining the Mayor's reputation and giving Shreck the opportunity to propose Penguin as a replacement. Batman confronts the Penguin, but Catwoman appears while firebombing Shreck's department store, and Penguin escapes. After a fight in which Batman knocks her off a building, Catwoman survives by landing in a truck full of kitty litter.
Catwoman agrees to help Penguin with a plan to ruin Batman's reputation by framing him for the abduction of Gotham's "Ice Princess" beauty queen. While traversing the rooftops to find the Ice Princess, Penguin's goons disassemble the Batmobile and plant a device into the car that will allow Penguin to control it. Distracted by Catwoman, Batman is unable to stop Penguin from murdering the Princess using a swarm of captive bats. When Catwoman rejects the Penguin's amorous advances, he responds by attacking her with his motorized helicopter umbrella. As the umbrella takes her up into the sky, Catwoman narrowly cheats death again as she falls into a rooftop greenhouse. Fleeing from the police, Batman realizes that the Penguin is remotely controlling the Batmobile, taking it on a rampage through Gotham. Batman disables the control device, but not before recording the Penguin's mocking insults about how gullible the people of Gotham are.
At a press conference for Penguin organized by Shreck, Batman broadcasts the recording, destroying the Penguin's public image. Enraged, the Penguin flees to the sewers and orders his gang to kidnap all of the first born sons of Gotham's citizens. At a masquerade ball hosted by Shreck, Bruce and Selina deduce each other's secret identities. The Penguin suddenly invades the party and reveals his intention to drown the kidnapped children, including Shreck's son Chip, in sewage water, prompting Shreck to offer himself instead. Batman defeats the kidnappers, forcing Penguin to unleash an army of penguin soldiers to destroy Gotham with missiles. Piloting the Batboat through the sewers, Batman redirects the penguins to instead fire on Penguin's hideout. The Penguin attacks Batman in a rage, but ends up falling through the ceiling of his lair and into the toxic water.
Catwoman ambushes Shreck in a vengeful attempt to kill him, but Batman stops her and unmasks himself. Before Bruce can reason with her, Shreck shoots him. He shoots Selina until he runs out of bullets, her now having only two lives left. Putting a taser to her lips, she kisses Shreck while grabbing hold of an exposed power cable, causing a fiery explosion. Bruce digs though the rubble but finds only Shreck's charred corpse. Mortally injured, the Penguin tries to take his umbrella to kill Batman, but collapses dead before he can finish. His penguin family carries out a makeshift funeral, pushing his body into the water.
Afterwards, as Alfred drives Bruce home, Bruce spots a shadow outside resembling Catwoman. He finds a stray black cat and decides to take it home. As he leaves, the Bat-Signal lights up in the night sky as Catwoman watches from afar.

Question: How does Catwoman kill Shrek?
Answer: Putting a taser to her lips and kisses Shrek while grabbing an exposed cable wire.
Alternative answer: electrocution