I’ve been swimming for years and many people ask me has my hair ever turned green. The answer is no. The only people who need to worry about that are people who have very light hair. I agree that the chlorine does tend to dry out your skin and hair if not treated. Now there are many products that help combat the chlorine. The main shampoo I use after being in the pool is Ultra Swim.
        The chemical chlorine tends to have a tight bond to skin too. They recommend that you shower before you enter the pool. This keeps the pool clean and helps the chlorine have a looser bond to the hair and skin. I will be honest and say I rarely do this, but I do tend to shower after my workout in the pool and my skin feels much less itchy if I shower after. Also they created this product called Swim Spray  that loosens the bonds of the chlorine and I will admit my skin does feel less itchy after using this product.  
       All that aside, if your skin still feels itchy, put some good lotion on it! Right now I use Eucerin daily replenishing lotion. It does a good job and makes my skin feel moisturized and soft. If you swim heavily like I did back in the day, you probably remember putting lotion and getting a small whiff of chlorine. It's not just you; it happens to everybody. 
      Something that I do want to add to this is that some people have minor chlorine allergies. I have never met someone with a very intense one but occasionally people come out with a small amount of hives or a few red patches on the body. They oddly have a runny nose as well. Chlorine tends to be the easiest and cheapest way to keep a pool clean and sanitary, however newer pools are exploring different way to keep their pool clean. There was one pool I swam in that had its primary filtration system run the water through a UV light. This UV light killed the bacteria making the pool much cleaner. They still made sure that chlorine was used, however the pool definitely had much less of a chlorine smell in the air.   
       All that being said these are things that are supposed to help you feel more comfortable and safe in the pool. Try one or all of those things and let me know how it goes.
Keep Calm and Swim On,
Coach Kelcey