How much material do you have? If they are grown the traditional UK way - planted about 6 inches deep and then "earthed up" in ridges twice as the leaves develop - the final depth of the seed potato will be about 12 to 15 inches below the soil surface. If they are planted shallower, you risk growing green potatoes because they are partly exposed to the light. The new tubers grow from the stem above the old one, not from the roots below it.
Note, contrary to popular belief, the green color is not poisonous. It is just chlorophyll, like any other green vegetable. But you won't want to eat green potatoes, because they contain a higher concentration of solanine, which makes them taste horribly bitter. Solanine is toxic, but an adult would have to be literally starving to death to eat enough green potatoes to get seriously ill, because of the taste. (Young children may be more sensitive to the toxin, of course)
Store bought potatoes may have been treated with growth retardant to stop them sprouting before being sold.
The most important reason for planting certified seed potatoes is to avoid the situation where the parent plants have been infected by virus diseases relatively late in their growth cycle, which doesn't have much visible impact on the plant or on the crop yield and quality, but which will remain in the tubers and infect the next generation right from the start of its life.
If you have power tools, I would just break up the turf and plant the potatoes straight into the ground. Put a layer of grass clippings one or two inches thick in the bottom of the planting trench and bury the seed potatoes in it. That will do more good that covering the ground surface with mulch. The potatoes should form a dense leaf canopy which will keep the light from any remaining grass and check its growth, but even if it does grow it won't harm the potato crop underneath the grass roots.
Potatoes grown the traditional way are an excellent method of getting rid of weeds, because the ground gets cultivated four times in their growing season - when they are planted, twice when they are earthed up, and again when they are harvested.