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:
*** Spoiler - click to reveal ***1. A mad dash from left/right to right/left. He will light up blue as he does this.
Jump up, double jump, and then use Square to Air Tackle him.
2. On either edge of the stage, he will be struck by lightning and start sending shockwaves towards you. The first few times he'll do this, early on in the fight, it will simply be ground shockwaves, but as he gets hit more, he'll include spiked shockwaves that arc overhead and alternate. After that, he will even increase the speed of the attack. In any form of this attack,
once he is done sending out his attacks, he will be visibly blue for you to go over and whack.
3. He'll jump into the background, or maybe the foreground (had this happen once), and throw rolling pins to either side of the stage. After, he will attempt to land on top of you, turning blue from the impact as his pins fly back towards him. Either take the time to whack him with your umbrella or Air Tackle him at this time.
4. He'll amass the Mafia lackeys into a large ball and start rolling towards you from either side.
The safest spot is directly under it, where there is a malformed gap in the ball's structure.
As it rolls away, use the same safe spot to rise up to the top and whack the Don.
5. He'll jump to the foreground and call for sandbags to fall down. You cannot harm him in this attack, but he can harm you. Early on, it'll just be sandbags to avoid, but midway through the fight, a Mafia goon will be dashing across as well, meaning you have to avoid two attacks in this attempt.
By delivering the killing blow (with everything turning red temporarily), the trophy should pop up.