He used what is called a torsion balance. His experimental method is outlined very nicely in this video.
After Coulomb published the result of his work, however, it was debated as to whether his experiment really did provide enough evidence to support his claim that the force between two point charges really did follow the equation we now call Coulomb's Law. This was because subsequent attempts to recreate his experiment proved that it was very difficult to obtain consistent results using Coulomb's method. If you are interested, here (DOI) is a very detailed report which outlines one of these recreations. This author comes to the conclusion that Coulomb's experiment was very very sensitive to any slight change: "Thus we see how just a single, slight, and easily invisible difference between Coulomb’s apt prescriptions and a replication may lead to entirely different results" (561).
Of course, it was later shown more solidly that Coulomb's law does hold up to experimental evidence, but it is still debated as to whether or not Coulomb jumped to conclusions, so to speak, and simply got lucky.