The simplest answer to this is to cook chicken thighs, not chicken breasts. When lean chicken breasts are initially cooked to a safe internal temperature of 160 degrees, they are at their optimal point for moist, tender meat. Because they have so little fat, when you re-heat the meat, you get rid of the little bit of liquid (essentially water, or meat juices) that is there–the result is dry chicken. (And you can't get around this by cooking the chicken to a lower temperature initially: doing so creates a breeding ground for bacteria).
There is an important exception: if the chicken breasts are braised (cooked in liquid), you are unlikely to perceive them as dry. Chicken stew, for example, reheats well, especially if the meat is cut in bite-size pieces
Chicken thighs, on the other hand, can be cooked and reheated because the fat, which does not cook out as easily as liquid, keeps them tender and juicy much longer.
The rule of thumb here is that the leaner the meat, the more easily it dries out.
See our collection of Chicken Recipesfor more ways to cook chicken.
Recipe: Roasted Chicken Thighs Provençal