I did the same thing as far as depth when I built my 4ft. x 12 ft. raised bed. I dug up the sod, removed it entirely, turned the ground with my shovel and broke it up with a hoe to a depth of about 6 inches. I got some top soil from a local distributor and hauled it home in my pickup truck, saving me at least a $50 delivery fee. I found the soil they sold to be a bit lacking so I put in a good bit of my compost to boost it a bit.
I am now getting some bermuda grass coming up in places but it is not enough to worry about. I do not use a fabric cover but plant directly in the soil and mulch once it is planted to keep weeds and help save on watering.