Only three containers of chlorine exist in the game. One can be purchased (or pickpocketed) from Ralph at Mick & Ralph's in Freeside, one is found in a storage room behind Mortimer's desk in the Ultra-Luxe, and the third is available from Blake at the Crimson Caravan Company.
The only use of chlorine is during the quest How Little We Know if you side with Nero and Big Sal by turning Cachino in. Big Sal asks you to see if or how you can help Clanden. If you pass a Science check, he tells you that the Ultra-Luxe uses chlorine in their pools and that you can either find it there or try to find their supplier. The chlorine is needed to make a chlorine bomb which is evidently used during the fight for Hoover Dam.