This does not have to be done in a single game. You must Checkmate the opponent using each of these pieces at least once: - Queen, Bishop, Knight, Rook and Pawn. It is rather easy to checkmate with these Major pieces as long as you retain them near the endgame while playing on Difficulty Level 0. If you have used any of these pieces to deliver a Checkmate in the Campaign and Standard Puzzles in Challenge Mode that will count as well.
For Checkmating with Pawns, what you want to do is to limit the Opposing King to 2 squares on the last file by controlling the rest of the board with a Queen and a Rook, 2 Rooks or any other possible combinations of the bigger pieces. You should have 2 pawns side by side on the same column of these 2 squares. Advance them all the way to the other end and you’ll checkmate the King.