Origins of the phrase "How killing!"
I found that the expression "killingly funny" (from which "How killing" comes from) is used to describe something that is farcical or screamingly funny and this term is related to or characteristic of the burlesque theater. The adverb "killingly" in fact, stands for "in a very humorous manner".
It’s first recorded (OED) as being used in this sense in 1844. It’s similar to the hyperbolic idea of dying with laughter. My own impression is that, at least in British English, it’s associated with middle and upper class speech.