How do you determine if a salt is behaving as an acid or base?

Whether the H^+ from water is pulled away or not.

NH4^+ + H2O ==> H3O^+ + NH3
F^- + HOH ==> HF + OH^-

BUT KCl, for example, will not hydrolyze at all.
K^+ + HOH ==> KOH + H^+ BUT both KOH and H^+ are strong base and acid and it will not hydrolyze. Note that with NH4^+ salts, you get at least ONE product that is weak. Same for HF, a weak acid.