Boil a large volume (at least a half liter) of chlorine bleach, just until crystals start to form. Do this outdoors or under a fume hood, to avoid inhaling the vapor. Boiling bleach disproportionates sodium hypochlorite into sodium chloride and sodium chlorate.
3 NaClO → 2NaCl + NaClO3


As soon as crystals start to form, remove the bleach from heat and allow it to cool.

In a separate container, prepare a saturated solution of potassium chloride by stirring potassium chloride into water until no more will dissolve.

Mix equal volumes of the boiled bleach solution and potassium chloride solution, taking care to keep solids from either solution out of the mixture. Potassium chlorate will precipitate out, leaving sodium chloride in solution.
KCl + NaClO3 → NaCl + KClO3


Cool the solution in the freezer to increase the potassium chlorate yield.

Filter the mixture through filter paper or a coffee filter. Keep the solid potassium chlorate; discard the sodium chloride solution.

Allow the potassium chlorate to dry before storing or using it. NurdRage has a video of the process, if you'd prefer to see how it's done.