A five-year investigation by Benjamin Radford, documented in his 2011 book Tracking the Chupacabra, concluded that the description given by the original eyewitness in Puerto Rico, Madelyne Tolentino, was based on the creature Sil in the 1995 science-fiction horror film Species. The alien creature Sil is nearly identical to Tolentino's chupacabra eyewitness account and she had seen the movie before her report: "It was a creature that looked like the chupacabra, with spines on its back and all... The resemblance to the chupacabra was really impressive", Tolentino reported. Radford revealed that Tolentino "believed that the creatures and events she saw in Species were happening in reality in Puerto Rico at the time"
The Chupacabra is extraterrestrial only in the sense that the first reporter imagined it based on a science fiction movie.
In late October 2010, University of Michigan biologist Barry O'Connor concluded that all the chupacabra reports in the United States were simply coyotes infected with the parasite Sarcoptes scabiei, whose symptoms would explain most of the features of the chupacabra: they would be left with little fur, thickened skin, and a rank odor.
According to biologists and wildlife management officials, the chupacabra is an urban legend.
So since they don't exist, they don't have a real origin, just in the minds of the people who believe in them.
All references from Wikipedia. Feel free to check their references.