Big squares of foam!
I laid out the fabric (three of them were made using old t-shirts) and cut it to size.
Then I laid the foam (flat side down) on the fabric, followed by the 3/4 inch plywood, and used my gun to staple the fabric to the wood.
Lastly, I added skinny wood bars to the bottom that will rest inside the milk crate to make the lid fit tight. These will keep the lid in place when you sit on the crate and provide it with a bit of extra strength. The brad nailer function of the gun made fast work of this step.