Just like in swimming pools, chlorine is what city systems use to clean their water, from time to time they'll even do a "chlorine flush" where they dump a ton in all at once to clear the pipes out. For most people it just means their tap water tastes more chlorinated than usual for a few days but if you're sensitive to chlorine, and many people with eczema are, it can cause all sorts of problems.
Some folks with eczema use a waterproof sunscreen lotion before going swimming in a chlorinated pool. It acts as a barrier between their skin and the chlorine.
I don't have eczema or dry skin, just hives, so I don't know if the meds would be the same. I've been on 3 or 4 different prescription allergy medications - I've had a hard time trying to find a doc who knows anything about physical allergies - none really worked. In a pinch I use OTC children's Benedryl because I can control the dose (it's a liquid) to avoid the
. Mostly I just avoid the allergen.