The basics of one arcade game is simply:
1. Defeat 4 enemies
2. Exit the area (you'll have to see the load screen).
(Some areas, like some of the doors in the main tent in Big Top, doesn't need these screens)
If you don't want to continously load the previous file without saving (see this solution at The Destroyer achievement), wait with this achievement until you get acess to act II
1. Pick easy difficulty. It's simply faster.
2. Pick one area you prefer in act I or II. I recommend Valley of Spirits since there are so many enemies that only need one hit there.
3. Play alone or with a (good) friend, so you'll know that you'll win 15 times in a row
4. The winner should take down four enemies. The other one doesn't have to do amything.
5. Exit the area
6. Either take take four enemies on that side of the exit or just return back.
7. If all enemies are defeated go to another area in the act you prefer.
if you want to continue in the same area, (especially after all enemies in all areas are defeated) visit the other act and enemies in the first act will respawn.
7. After 60 enemies you should be done.
Some guides mentions that this has to be done in one sitting, but I did this in several. Just I know this because it popped up before the 20-times achievements, even if I first did 10 one day and 5 the next. Just remember to let one controller win all the time.
If you visit all areas in Atlantis enemies will respawn without you having to switch acts. Some of the enemies there need special finishing moves, so the area can be a bit frustrating.