How do you get the card key in Pokemon Blue?

Since a detailed answer would take much too long, I'm simply going to say this:

A good rule of thumb to find the Card Key is to talk to everyone you've beaten after the battle. One guy in particular (on one of the upper floors, standing next to a table) will say, "Oh, no! I've dropped the Card Key!"

A Poke ball will then appear next to him. Inside is the Card Key.