Here is my achievement guide, for every single achievement in Turnip Boy Commits Tax Evasion.
All of the achievements are ordered and have chapters attached in the video.
For this achievement, you will need to basically do all the other achievements and obtain the "Anarchist" achievement for destroying every single tax document. Once you do that, you can fight mayor onion (even if you have already), and after defeating him regularly, you'll have a secondary boss fight that unlocks the achievement.
It's fairly simple. Go to the right side of the map, kick a bomb at the cherry, he'll attach the mayor.
Do the same on the left side of the map with another cherry.
Return to the middle of the map, and do the same with the original cherry. Boss defeated.