This achievement is fairly straightforward to unlock once you understand the requirements.
On the official boards they clarified the achievement requirements somewhat in that you have to declare an attack on each opponent, the CPUs cannot kill each other without your intervention.
KMetalmind and JORAX further clarified in the comments below that the CPUs must be eliminated by you. The CPU cannot be eliminated by suicide (such as his last harvester falling into the mist) or by other CPU players.
You will need to set up a custom game, 4 players, using skilled or expert AI opponents. It may help to choose opponents all at the same difficulty, as that way they will react the same way making this achievement a little more controllable.
For map, I chose Morana Base (which is the map that is used as the first map in the Empire Campaign). In it, each player starts out with a cannon which comes in quite handy for this achievement.
On this map the CPU opponents will generally place harvesters such that they soon end up on islands with nothing but the cannon they started with and 2 or 3 additional territories. You should do the same. I carefully placed my harvesters so that I was left with 4 territories - eliminating the possibility of a 3 carrier invasion.
What you want to be doing is building up a large supply of walkers and place at least one ammo into your cannon. The CPUs will eventually get into a 'cold war' of carriers. You will be a bit behind them because of your troop buildup as well as buying the ammo - that is okay. When possible you want to build up carriers as well.
When you see that a CPU is likely to be eliminated - if he has only two territories left, for example - then use your cannon ammo to shoot the CPU who is likely to die. This will declare an attack on him per the achievement requirement. As long as you do not shoot or kill units from any other player before this declared opponent is eliminated, you are in the clear. Other players can attack this CPU but based on comments below it appears you have to be the player that eliminates the opponent.
Given this, if you see an opening for a kill - go for it. In my game at one point I had two carriers and one cannon ammo. I shot the cannon at a CPU's lone cracked territory putting it in the mist, then I used my two carriers to take over his last two spots. For my trouble I received his two carriers letting me keep up the press on the remaining two players.
Repeat this for the next opponent. When it is down to you and one other CPU, you can eliminate him however you choose.
Note, you are allowed to defend yourself against enemy attacks however on this map, that means they have flown over to your land and will likely result in a loss without you being allowed to attack the invader.