The earliest printed reference to the origin of this proverb is attributed to British playwright Ben Jonson in his 1598 play, Every Man in His Humour (performed first by British playwright William Shakespeare).
...Helter skelter, hang sorrow, care will kill a cat, up-tails all, and a pox on the hangman.
In this instance, "care" was defined as "worry" or "sorrow."
Shakespeare used a similar quote in his circa 1599 play, Much Ado About Nothing:
“ What, courage man! what though care killed a cat, thou hast mettle enough in thee to kill care. ”