You need the oxidation number for Cl, so we will use only the ClO3 1-.
Oxygen is not in a peroxide. It is in "chlorate" so the oxidation number of oxygen is 2-. There are 3 oxygens in the ion. 2- x 3 = 6- total. The chlorate ion has a 1- charge so there are 5- charges to be balanced out by positive charges. That means that the Cl (the only other atom there) has to balance out the 5- left over by having a 5 + oxidation number.
(Cl 5-) + (O3 6-) = ClO3 1-
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