This achievement is possible to obtain using system link and no extra Xbox Live Gold or Silver accounts other than your own.
You need two xboxs, 2 copies of the game, and 5 controllers.
One digital copy of the game counts as two copies as long as you have access to the account that purchased the game and the console the digital title is registered to. The account that bought the game has to be used on the opposite console to download the second copy and has to be signed into Xbox Live on that console in order for the game to launch. Otherwise you need two physical copies or a combination of both. I used a digital copy shared between two xboxes.
If you don't have 5 controllers, Rock Band/Guitar Hero instraments work. I used 4 controllers and a Guitar Hero guitar.
The game only allows 4-player splitscreen which is why you need a second xbox as the achievement requires 5 different accounts in the lobby. Load up both copies of the game. Go to Multiplayer. Then choose System Link.
The next step is where you are going to sign in 5 gamertags, one of which is your own. (4 on one xbox and 1 on the other or any combination depending on what type of controllers being used.) These are 5 unique gamertags, NOT guest accounts. You can create the 4 extra accounts before loading up the game, or they can be created in-game on the Identity sign-in screen. I created mine in-game. Hit the Xbox Guide button on the controller not signed in. Go down to Create Profile. Type in a name. Pick an avatar. Then choose "Done". You do NOT need them to be Xbox Live profiles. Repeat for the other 3 controllers. (or less in you have other profiles already that can be signed in)
Ready-up the profiles. Have one account host a lobby. I used the settings in BATTLE FIELDS's solution.
Mode: Team Elimination
Type: Last Man Standing
Under "Advanced Settings", look at the first two options. Make sure to allow for 5 or more players in the game, and then "Method for Player Assignment" should be set to "Player Control" so that teams are not locked to allow you to have your account on one team, while the other 4 are on the opposite team. All other settings do not matter.
Create the lobby and have the account(s) on the other xbox join. Swap accounts to the appropriate winning and loosing teams. Launch the game. Have one account on the loosing team shoot and kill the other three and then use a grenade to kill himself. Do this 5 times on each of the 10 maps.
The 10 maps in order in the game:
The achievement will pop without the need for any other Xbox Live enabled gamertags, other than your own. I got this achievement using this method. I used my Gold account, my dummy account which is Silver, and 3 other non-Live accounts created using the method I described above.