You don't need a lot of space to start growing your own potatoes. In fact you don't even need to have a potato tub. The only thing you need to succeed with growing potatoes is a large bag of compost.
First you need to get some seed potatoes. You can use the potatoes you purchase with your grocery shopping but these are not disease resistant, so it is much better to by proper seed potatoes from your local garden shop.
You will need to empty the bag of compost. When you do this make sure that you don't rip the bag, as you will be using it to grow your potatoes. Cut the top open carefully and then empty the bag; try to empty the bag somewhere suitable, as you will need to store this compost until you need it later.
You then need to roll the top of the compost bag down. This can be tricky as some bags that hold compost are very thick. You don't have to roll it all the way down, about half way is fine. Put a small layer of compost in the bottom of the bag, and place your seed potatoes on top. You won't need a huge amount of these seed potatoes as only 2 or 3 will produce you a healthy crop in you bag, so use your best judgment.
Then cover the seed potatoes with a few inches of compost and water in. That is all you need to do to start growing your potatoes.
After a few days or a week (depending on the weather), you will see shoots coming up out of the compost. When this happens you need to cover them with more compost, and unroll the bag slightly.
You need to keep adding the compost every time you see the shoots, until you have filled the bag back up and you can fit no more compost in the bag. It is then time to leave the potatoes to grow.
The biggest problem you will face with any type of container gardening such as this, is letting the compost dry out. In a container it will dry out much quicker than if it was in the ground. So try to water your potatoes daily and keep a close eye on them. Potatoes need plenty of water to swell and grow in to healthy tubers.
The potato plants will grow and mature out of the top of the bag, sometimes they can get quite big, so make sure you can support them with something, or lean the bag against at wall. When the potato plants have finished flowering you can empty the bag and have lots of new (baby) potatoes for cooking. Or if you prefer larger potatoes you can leave them in the bag for a few weeks longer.
Once you have emptied the bag, you can spread the compost on your vegetable patch or on your flower board. But try to avoid using the same compost for more potatoes as this may cause a potato disease to spread.