You will get helmets every time you destroy an object or defeat a Roman outside of the Olympic events. As you fight Romans they will drop 'x3' and 'x10' multipliers. Pick these up to increase the number of helmets you receive. After picking one of these multipliers up, you will only have a limited amount of time before it will go away. If you quickly pick up another to add to your multiplier, you can extend the time that it lasts. In big fights you can increase your multiplier to well over 'x100'.
Playing through the game normally you would likely never get near this many helmets (I got roughly 15,000 when I played the game), so you'll have to grind for them. This is best done in the same place as the “100 combo hits achieved” achievement, except it cannot be done in the King of the Arena event.
These achievements are for total helmets acquired not how many you currently have, so feel free to spend your helmets for things like upgrades and health replenishment.