Given that, $f(x) = x^{2}.$

Now, $f\left(f\left(f(x)\right)\right) = f\left(f\left(x^{2}\right)\right) = f\left(x^{4}\right) = x^{8}$

$\therefore \dfrac{f\left(f\left(f(x)\right)\right) }{f(x)} = \dfrac{x^{8}}{x^{2}} = x^{6} = (x^{2})^{3} = (f(x))^{3}.$

So, the correct answer is $(C).$