First: Buy the One-Hit Hero Badge from the Badge Seller for 500 Pons.
Next: Decide the boss that you wish to take on. I'd recommend the first, the Mafia Don, of Chapter One.
After making your way through the hideout and coming across the stretch, make sure you pause before encountering him to make sure the badge is equipped. Even though it will still show you with four energy, you will only have one chance before you will die if you are hit.
The Don has a simple attack pattern, classified in five attacks. They are:
By delivering the killing blow (with everything turning red temporarily), the trophy should pop up.