Janet's basically put it in a nutshell.

The lithium chlorine is priced in "Boo-Coo" amounts--far beyond the amount of chlorine it delivers. Effects of high lithium levels in pool water are not really known around here--I don't know how you test for it. Potassium based chlorine is TOTALLY new to me--I'm guessing, like lithium, it's an attempt to bypass the issues of calcium based chlorine. Again, what do high levels do? Both lithium and potassium are metals, like sodium and calcium---and copper.

Bleach is cheap and totally side-effect free. Whether it's bleach or liquid chlorine it's the same stuff. Sometimes the L/C is higher concentrations of Sodium Hypochlorite, but you just use half the amount for the same effect.