Horseradish is a very strong perennial plant, it doesn’t need much or any attention. The leaves will die off in winter but in spring it will grow back in.
Horseradish seeds are sterile, therefore you would use the roots to propagate it. Apparently best are the thin ones 6-12 mm, but for me any root will do. The roots are best planted 50cm apart and 5cm deep. Replanting should be done in spring or autumn.
Horseradish will take care of itself, you can grow it on the same spot for quite a few years, it will even propagate itself with roots and can be sometimes invasive. Despite of this, it is a good idea to replant it once in few years, because it has for some reason a tendency to gradually die off when grown in the same spot for many years. Professional horseradish growers in Slovakia do harvest it in autumn and replant every year for better yields, but if you just grow it for your own use, you don’t need to worry about that. By the way horseradish is hardy enough even in Slovakia to overwinter without any problems.
The “strength” of horseradish depends on the weather, if you harvest it early in autumn it won’t be as strong, but harvested once the cold weather starts killing off the leaves, it will be at it’s best. To harvest it, just dig out the root, you can use it whole or replant a piece and use the rest. Normally I don’t worry about propagating it as it is quite invasive, just digging out the whole root. Even so, if I do accidentally leave a small piece of root in the soil it will grow back.