Annie, before I got my allotment I used to grow Sharpes Express, an early potato in black plastic dustbins with a few holes drilled or melted in the bottom.
A few inches of ordinary compost in the bottom, tubers on top and then about 6 inches of compost on top. Every time plants got about 6 inches high I added more until just the top two leaves showing.
Crop not as big as you get in the ground but just as tasty. Be careful not to overwater though or they go soggy and mouldy.
Good luck!