How to purify water with activated carbon?
The main reason activated carbon is used in water treatment is that its large surface area is useful for adsorbing volatile organic contaminants and chlorine. It is not effective at removing inorganic ions (dissolved minerals, including fluoride), but these are removed quite well by distillation. I think distillation first would make more sense—distillation will remove less volatile contaminants so that less has to be adsorbed by the carbon.

Tap water has to meet certain standards (that depend on your area) for dissolved minerals and volatile organic compounds, so depending on the end use of the water, removing the small amount of remaining contaminants may not be necessary. Formerly, double-distilled water was the standard for making highly purified water, but certain applications are more or less tolerant of certain types of contaminants, and double-distillation requires a lot of energy, so it has largely been replaced for many purposes.

Consumer water filters usually contain filters to remove particulates, ion exchange resin to remove inorganic ions, and activated carbon to improve the taste of the water, and as far as purification for laboratory use, combinations of filtration, deionization, reverse osmosis, ultrafiltration, etc. are common these days. Different processes are better at removing some types of contaminants than others, so combinations are used. Distillation of tap water followed by treatment by activated carbon will produce quite pure water by consumer standards, but might not be pure enough for say semiconductor microfabrication or biomedical applications.