Bonus levels are generally hidden inside enclosed areas (areas that are completely blocked off by enemies, bricks, and boulders), that usually require pickaxes or dynamite to get into. You will know you find one when you find a golden staircase that isn't at the end of the level.
Once you get into a bonus level, you have 15 seconds to collect as much loot as you can. This achievement is cumulative, so you don't need to collect all 1,000 gold in a single bonus level (which isn't really all that hard any way in later levels, as more loot shows up the farther into the game you are).
If you happen to trip a trap during a bonus level it will kick you out of it, so do be careful and make sure your character never has to go through any dirt tiles to get to where you clicked. All of the dirt tiles that don't have any traps around them will go away when you get close to them, as they do in normal gameplay.
You should be able to get this within 2 or 3 bonus levels, and may even get it in a single level if you get to a bonus level that's late enough in the game.
*Keep in mind that there is a 15 second ad you must watch every single time you want to enter a bonus level. AFAIK there's no way around this.