Well, if we apply

Gauss' law we get:

where Φ

_{S} is the flux of the electric field E through the surface S. In this case, as the direction of the field is always normal to the sphere's surface we can write Φ

_{S}=ES, so we get:

As the electric field is F/q we have, at last (q

_{2} is the charge we put in the electric field generated by the charge q

_{1}):

A more straightforward way to get the inverse square law would be to derive your expression (using r as the independent variable), as E is also dV/dr (where my V is your E, the electric potential):

(the "-" only means that the direction of the force vector is opposite to the increase in the electric potential)

Hope it's all clear now... what you did is also a common way to verify Gauss' law with an easy example...