The phrase, with variants, predates Batman: The Animated Series (1992 to 1995).
The earliest result in Google Books is a snippet of Me, Minsky & Max by Bruce Pollock (Page 221, 1978):
This whole dumb trip might have been for nothing! Zero, zilch, zip, nada, nothing. I wondered how I'd recognize him. Would he give off some kind of winning quintessential Stillman gleam, or would he come laden with the aroma of a loser?
The earliest full view result is from InfoWorld magazine (7 Feb 1983 - Page 20, Vol. 5, No. 6):
You've obtained your first home computer, taklen it fresh out of the box, set it up on the computer stand and plugged it in. Nothing happens. You click a few switches, tinker with the keys and try again. Zilch, nada, nil; zero response. The box is declared dead on arrival.
As FumbleFingers commented, National Lampoon (Page 311 - 1974 - Snippet view) has an earlier variant without nada:
Sooner or later the whole damn world's going to know, anyway. The NatLamp's washed up. That's right. We're finished. Fresh out of ideas. Empty. Barren. Bumed out. And there's nothing left. Zip. Zilch. Zero. The square root of sweet fuck all.
The repetition is for emphasis and to make a point, and they have a jocular effect which is reminiscent of Monty Python's dead parrot sketch from 1969:
'E's not pinin'! 'E's passed on! This parrot is no more! He has ceased to be! 'E's expired and gone to meet 'is maker! 'E's a stiff! Bereft of life, 'e rests in peace! If you hadn't nailed 'im to the perch 'e'd be pushing up the daisies! 'Is metabolic processes are now 'istory! 'E's off the twig! 'E's kicked the bucket, 'e's shuffled off 'is mortal coil, run down the curtain and joined the bleedin' choir invisible!! THIS IS AN EX-PARROT!!
I also joked this is similar to Roget's Thesaurus so here's an entry from an 1879 edition: