This process is a valid (albeit dangerous) way to make potassium chlorate. The process can be summarized by the following reactions:
Step 1: When bleach (sodium hypochlorite) is heated, it decomposes into sodium chlorate and sodium chloride:
3NaOCl --(heat)----> NaClO3 + 2NaCl
Step 2: The potassium chloride displaces the sodium in the sodium chloride to form potassium chloride and sodium chloride:
NaClO3 + KCl -----> KClO3 + NaCl
The problem with this synthesis method is that an excessive amount of stable table salt is produced. Not only does this reduce the effective yield of desired product, it requires purification, which is extremely dangerous for chlorates. You can lower the amount of salt produced by not adding the potassium chloride. By doing this, you will be left with SODIUM chlorate. Sodium chlorate still works just as well as potassium chlorate. The process will still be extremely dangerous, but the risk is lessened.
A safer way to make sodium chlorate would be to use the make your own bleach method, but heat the pot instead of cool it. Still I do not recommend anyone other than a qualified chemist to carry out anything involving chlorates.