This will be one of the last achievements you unlock and it will not come through natural story progression at least without the DLC. I think i was somewhere around 400 meals delivered after I had completed the base game campaign without replaying any levels. The fastest way to grind this is to go to Story mode and switch it to practice. You can see the game mode in the top right corner and you can change it with Y on the controller or some button(?) on the keyboard. In practice mode you can just keep playing until you get the cheevo or get tired and want to continue another time. Choose level 1-1 as it only has one-ingredient dishes. Once you get the hang of it, you can watch TV or at least listen to something at the same time. The achievement will pop in the level, no need to back out.
I never timed it but I think it's possible to deliver 8-10 meals per minute.
In short:
Practice mode
Level 1-1
Chop, serve, deliver (I'm yours!)