Generally speaking, your eyes getting a bit of chlorine most happens when you are swimming in a swimming pool. The pool managers often use chlorine to disinfect the water, because its strong oxidation will kill the bacteria in the water. It is not dangerous and all you need to do is waiting and your eyes will recover itself in most time.
Well, generally speaking, when you have chlorine eyes, and you ignore it, your eyes will be painful and burning. And that can be annoying and burning. At this moment, you may need to try some ways to treat it. And according to some researches, milk can be very beneficial to chlorine in the eyes, for milk contains vitamin and many other healthy ingredients. So in my opinion, you can just apply some milk on your eyes in several days. Anyway, just be careful about it.
Oh, it seems that you have got something wrong into your eyes. Anyway, small amount of chlorine would not be serious because the irritative feeling would results in some tears, which would make the chlorine diluted and disappear. However, a lot of chlorine would be dangerous. I suggest you try to apply some eye drops after you flush your eyes with clean water.