Whilst this can be done against one opponent, I suggest you select three.
At the bottom of the screen during a match, you can see whether your opponents have any gold or carriers. If they have a carrier, you will see a propeller above their name. Build a cannon and shoot an enemy tile which has enemies on it (no buildings). The idea is to shoot it down to level one. The enemy units will usually move from the tile, as they do not want to go down with it. Once you have an enemy tile down to level one and vacant, attack their other tiles with walkers. If you try a harvester attack, make sure the level one tile is not connected to the tile the harvester is on.
When you eliminate the opponent by capturing/destroying all of their buildings and units, you will see a pop-up above their name. It will show what you received from them. You need to see at least a "1" next to every item to be awarded the achievement.