You can get this on one console with 2 PSN accounts and they need to be friends with each other. Make sure both accounts are signed up to PSN.
As PSN is free to sign up for and play online unlike the 360, there should be no need for any additional subscriptions.
I set up the game on my main account, from the main menu select Play By Mail.
You need to select an empty slot and pick your second account. You will then set up the game.
Pick Black as your colour and send the request.
You will then need to sign in to your second account on the XMB, this will quit you out of the game. Then reload the game.
Select Play by Mail and press Square then select the message.
You will now play the first move as White.
Play the f-pawn to f4
Make sure you send the message to your main account.
Now sign back into your main account (repeating the above steps for each move)
The game will play out like this.
Black (main account): e pawn to e5
White (second account): g pawn to g4
Black (main account): Queen to h4 (winning the game)
This is the shortest possible game of chess and on completing that move and sending the message, you will get the trophy.