From a poster (thanks Bree) on one of my listservs - the simple explanation and then the detailed explanation with link to site.
We have been growing potatoes in containers for years and it is really easy. You need to put drainage holes in the bottom and broken clay pot pieces to help drain water off the roots to prevent root rot. Add some rich soil (we compost all left over veggies from the kitchen) then plant your potato eyes. As the vines grow cover them with more loose compost and they will keep growing. Web site that explains the process in more detail.
Link to detailed instructions;
How to Grow Potatoes in a Container (Ciscoe's Secrets)
Get a clean garbage can or similar container. Plastic works great because it won't rust out. Drainage is absolutely necessary. Drill several 1/2 inch holes in the bottom. It also helps to drill some holes in the side about half-inch up from the bottom of the container.
Fill the container with about 6 inches of good potting soil. Mix in about a handful of osmocote 14-14-14 fertilizer. Osmocote is a slow release fertilizer that will stay active for approximately 2 1/2 months. Organic fertilizers formulated for acid loving plants such as rhododendrons also works well. (Note: After 2 1/2 months with osmocote, or about 1 1/2 months with organic, fertilize with a good water soluble fertilizer such as Miracle Grow about every two weeks according to directions on the label). Place whole seed potatoes in the soil. There should be about 5 inches between potatoes. Cover with an additional inch or so of soil. All potatoes should be completely covered with soil. Water the spuds in.
The potatoes will begin to grow. When the vines reach 4 inches, cover all but 1 inch with compost or straw. I like to use compost, because it is easy to reach in to pick potatoes. Every time the vines grow another 4 inches, keep covering all but the top inch. Eventually, the vines will grow out of the top of the container. It is a good idea to stake up the vines so they don't fall over and brake. Place 4 bamboo or wood stakes (one in each corner) and tie the vines to the stakes with twine. By now the whole container will be filled with compost. Soon the vines will flower. Not long after that, the vines will begin to produce potatoes all along the vines that are covered with compost in the container. Once they have become big enough, you can reach in and pick a few for dinner any time you want. These spuds are called "new potatoes." They won't keep long in the fridge, so pick-em and eat em. After the vines die back at the end of summer, the potatoes remaining are storing potatoes. You can harvest and store them as you normally would. These will keep well as long as they are stored in a dark, cool, and relatively dry location.
One last note: take care to provide adequate water. You don't want to drown the plants but it's also important the soil at the bottom never dries out. In late summer spuds may need to be watered on a daily basis. Use a watering can to water to avoid wetting the foliage. I found that keeping the containers in an area with morning sun exposure prevents the soil from drying out too rapidly and still allows enough sun for a bumper crop.
This method of growing spuds is really fun. You get lots of them without using much space, and it amazes visitors to your yard. Last year I harvested 35 large Yukon Gold and 55 good sized Peruvian Blue potatoes. Great served with brussels sprouts!