I live on Long Island in New York State. We have relatively mild winters, though we do get plenty of snow.
I cover my vinyl lined pool, but, to save the need to add water during the winter, I just "drain" the water that accumulates on the cover, back into the pool.
I have done this for at least ten years. The result is dirty water when I open the pool, but with a heavy shock and then 24-48 hours of filtering and it becomes crystal clear.
As such, is it even necessary to add chemicals before closing? If so, what is the feeling about using liquid clothing bleach (ex Clorox) instead of pool store chemicals?
Thanks for the help.

If you read around this site you'll soon learn that using bleach is the preferred method of, not only closing but, regular maintenance of our pools. If you'll close it according to the directions found here you won't have to shock it next year to open it.