NaHclO3 breaks into Na+ and HClO3-, and that further breaks into H+ and ClO3-, is that right?

I don't see a question here.

Is that the right way to break down NaHClO3?

It COULD be right but as far as I know that is no such compound as NaHClO3.

NaClO3 is sodium chlorate.
HClO3 is chloric acid.
I don't know of anything with the formula NaHClO3.

Yes, your statement is correct. NaHClO3, also known as sodium chlorate (IUPAC name), breaks down or dissociates in water into Na+ (sodium ion) and HClO3- (chlorate ion). The chlorate ion, HClO3-, can further dissociate into H+ (hydrogen ion) and ClO3- (chlorate ion). This process is referred to as ionization or ion dissociation, where a compound separates into its constituent ions in a solution.

To confirm such dissociation, you can consult reliable sources such as chemical databases, textbooks, or scientific literature. These resources typically provide detailed information on the composition and behavior of chemical compounds.