I got this achievement in Classic Skirmish, it cannot be done in Head to Head. Set the difficulty to 0 to make this easiest.
Play the first couple of moves until you get told "Opening Book mode ended, you're on your own".
Then use RB to get hints on how to play the rest of the moves. Use the hints which allow you to capture your opponents key pieces. Ignore pawns after you capture a few pawns because these will be key to getting the achievement later.
Try and get your opponents Queen and Rooks off the board before attempting this ideally while leaving either a Queen or Rook in play on your side, if you could get a Bishop to remain aswell then this will help even more.
Now when your opponent is down to just King and Pawns, this is where you balance out the piece amounts such that you have 2 less pieces than you opponent hence why it was important to keep the pawns in play.
You can start balancing out before you get your opponent to King and Pawns if you prefer but it's easier now because you'll have more control over how the game plays out as the pawns can't do much (unless they are in an empty column and none of your pawns are in the columns adjacent!) and the enemy King will probably have to do most of the capturing.
Once the pieces are balanced out and you have 2 less than the enemy, go ahead and go for Checkmate. But be careful not to capture any enemy pieces and undo your work! If you have to capture a piece then make sure the enemy gets one of your pieces in return.
Best of luck!