We need to remove the envelope
so we can modify the fills. Select the enveloped checkerboard, then select Convert to Editable Shapes from the Arrange menu. Ungroup (Ctrl - U) the checkerboard twice.
Select the large square and apply a Linear Fill using the two shades of black as shown.
Select the checkers and apply a Linear Fill using the two shades of black as shown.
Making the light and dark colors closer in value as the go to the back enhances the feeling of depth.