Note: Tracking the origins for the majority of popular idioms isn't easy, in fact, it's often impossible. Usually what you will find are plausible theories that try to explain a phrase's origin, but remember, they are just theories.
The quotes on this website come from older books, poems, newspapers, etc. They can be used as an indicator to determine how old some of these sayings are. Keep in mind, however, that if a saying is being used in an old newspaper, it's probably already well-known, so it's likely older than that.
knowyourphrase.com - Common Phrase Origins - Sayings and Their Meanings
There is an older form of this expression that goes: "Care killed the cat." The word
'care,' in this case, seems to be defined as "worry" or "sorrow." This form of the
expression goes back to at least the 16th century. It's used by a few playwrights
during that time. For example, an English playwright named Ben Jonson is said to
have used it in a play called Every Man in His Humour, 1598. Another playwright
named William Shakespeare used the expression in the play Much Ado About Nothing, which is thought to have been written in 1599:
"What, courage man! what though care killed a cat, thou hast mettle enough in thee to kill care."
The earliest that I could find of the modern phrase being used in writing is from a book by James Allan Mair called Proverbs and Family Mottoes, 1891, where it's simply listed as a proverb on a page:
"Curiosity killed the cat."
* My boss warned me that curiousity killed the cat after I kept pestering him to tell me why he fired his last employee.