The order of basic strength of ${\mathrm{ClO}}^{-}$, $\mathrm{Cl}{{\mathrm{O}}_2}^{-}$, $\text{Cl}{{\text{O}}_3}^{-}$,   $\mathrm{Cl}{{\mathrm{O}}_4}^{-}$ is       

If the acid is strong its conjugate base is weak. Same like if the base is strong its conjugate acid is weak. Among the given option $\mathrm{Cl}{{\mathrm{O}}_4}^{-}$ is the conjugate base of strong acid  ${\text{HClO}}_4$, therefore basic strength of $\mathrm{Cl}{{\mathrm{O}}_4}^{-}$ is weak. Among the given option ${\mathrm{ClO}}^{-}$ is the conjugate base of weak acid    $\mathrm{HClO}$, therefore basic strength of ${\mathrm{ClO}}^{-}$ is strong.       

Hence the correct option is (C).