I have grown a variety of potatoes in pots for about twenty years - probably a hundred sorts so far.
It is not difficult, and the instructions below will enable you to do it, too, with minimal expense.
Forget about potato barrels and other expensive stuff. All you need are some builders' buckets which cost about £1 each, and a few potatoes which have begun to sprout.
First, take your bucket, and drill some holes in the bottom. Place the bucket on a thick board before drilling. If you don't, the bucket will probably split as the drill goes through. You need plenty of holes, so that water can drain out.
Here it is, safely drilled and ready for use.
Now take one very large seed tuber, or two medium ones, or three the size of hen's eggs, or four very tiny ones. Put a two inch layer of compost in the bucket, then place the tubers as far apart as possible and near the edge, and cover with compost so the bucket is about one third full.
Water it a little to keep the compost damp. Put the bucket where it will be rained on, and in plenty of light. When the leaves start to push through, keep topping up with compost, until the bucket is full.
When the stems develop, you will need to water very regularly - probably once a day. Do not over-water. Better to be slightly dry than over-wet. You judge the need for watering by picking up the pot and feeling the weight.
Potatoes in pots often develop more quickly than those in the gound. Some varieties will be ready after 10-12 weeks; others (maincroops, like Golden Wonder or Sarpo Mira) may take 16-18 weeks.
Here's what you pot should look like, after 2-3 months:
When they are nearly ready, you'll notice the leaves start to go slightly yellow:
You can either leave them in a little bit longer, or take them out, wash out the container, and put in your next lot of tubers. Repeat sowing can be worthwhile right up until the beginning of August.
Other containers are possible. Florist's buckets are fine, or beer brewing containers sawn in half, or 5 gallon drums sawn in half longways or across. Certain varieties do extremely well in trays, shown below:
On the next page I have listed the varieties which I have found perform best in different kinds of container.
You might also consider polypots. These are quite cheap and sold by JBS seeds. However avoid big containers (dustbins, large barrels, enormous planters) ; the yield of potato you get doesn't justify the amount of compost you have to buy.
I haven't mentioned this yet, but don't use soil in the containers. It will set like concrete and the yield of spuds will be virtually zero.
One other thing - with reasonable compost, you do not need any form of fertiliser when growing potatoes in pots. It makes no difference at all to the yield. Potatoes need water and light.