You placed salt on the surface of the chicken just before cooking, and then noted that it was salty on the surface but not deep inside. That's to be expected. If you left the salt on for a long time, it would start to eventually react and interact with the tissues deeper in the chicken, but that would be more like a dry cure, which draws moisture from the meat (think like a European ham or a USA "country" ham). But the dry cure takes a long time to dry out meat, so possibly just leaving it on longer will at least have some movement of the salt into the tissues via osmosis. Still, I think a more reliable method is delivering the salt along with water, because the water also gets delivered into the tissues.
That's why brining is the method to use if you want to infuse some of that salt and more moisture into a chicken, but it requires time to do its work.
How does brining work? Brining promotes a change in the structure of the proteins in the muscle. The salt causes the protein strands to become denatured, or unwound. This is the same process that occurs when proteings are exposed to heat, acid, or alcohol. When the protein strands unwind, they get tangled up with one another, forming a matrix that traps water.
In most cases, we add sugar to the brine. Sugar has little if any effect on the texture of the meant, but it does add flavor and promotes better browning of the skin
Cooks' Illustrated: The Science of Brining
For a whole chicken, they recommend a brine of 2 quarts of cold water with 1/2 cup of table salt for 1/2 hour to 1 hour. I've seen other articles they've published where they'd also have about 1/2 cup of sugar, as well. Yet others have a longer brining time, but a less intense solution. This one, at least, is a faster one, as far as the time you need to brine, vs the one I usually use.
Cooks' Illustrated: Brining Meat