Chlorine is necessary in growth for plants. You are correct in thinking about it as a separate cycle. In this cycle, plants get chlorine from the ground through their roots: chloride ions are naturally occurring, and specialized cells in the roots uptake this chlorine for use in cell cycles. (I didn't know this until today, that was a really cool question.)

Chloride does not evaporate; it is left behind in the form of salt. Most salt in nature is sodium chloride, although potassium chloride and calcium chloride are also fairly common chlorine-containing natural salts.

Most chloride salts, including all of the common ones, dissolve in water very well. Plants get their nutrients, including chloride ions, from the water they draw up from the ground.

When you sweat and the water evaporates the Chloride ions actually are left behind. That's why you feel crusty and salty after your sweat evaporates. In terms of plants there are some small uses for Chlorine atoms. It's mostly toxic and plants have evolved ways to exclude it from tissues. Plants do need small amounts of chlorine for growth, however. If you deprive a plant of any chlorine, the leaves will turn yellow and die.