Klein provides two possible answers. The one he feels is more probable, is that the two words come from different languages, and just happen to have opposite meanings.
The Biblical קלס is related to Ugaritic qls, and is of Semitic origin. He writes that the original meaning of this base was "to stamp with one's feet".
The Post-Biblical קלס, according to this theory, derives from the Greek kalos (or kallos) meaning "beautiful". From this Greek root we get such words as calligraphy (beautiful writing), kaleidoscope (observer of beautiful forms), and calisthenics (we discussed the word here earlier.)
However, Klein does mention a less probable theory:
According to some scholars, the bases קלס (to mock) and קלס (to praise) are derivatively identical with קלס (to stamp with one's feet). The original meaning 'to stamp with one's feet', would have developed into 'to mock, scorn, scoff, deride' on the one hand, and 'to praise laud' on the other.(This isn't related to language, but the dual nature of praise can be seen in education and psychology as well. For more on this, I recommend my father's article Are Your Students Addicted to Praise?, and Alfie Kohn's book Punished by Rewards.)
Ben-Yehuda mentions another possibility, that the meaning of קלס - "to stamp with one's feet" may be derived from the Latin calx meaning "heel" and calciare "to stamp with the heels, tread". From this Latin root we get such English words as recalcitrant and inculcate.
One phrase that may or may not be related to the above is the גדי מקולס - the mekulas goat mentioned in the Mishna of Beitza 2:7. Kehati provides two explanations:
And one may prepare a kid mekulas on the eve of Passover - a kid roasted whole, including the head, with its legs and with its entrails, with the innards suspended outside the kid, the way the Passover sacrifice was roasted in Jerusalem (Rambam explains the term mekulas as "presentable," while others explain it as meaning a helmet, because the innards were suspended like a type of hat on the carcass, so that it looked like a mighty warrior carrying his weapons, as, for example, in regard to Goliath, who wore a copper helmet - the Targum of Goliath's "copper helmet" (I Samuel 17:5) is kulas dinhash)The Rambam, therefore, connects mekulas with kilus - praise, whereas Rashi does not. According to Steinsaltz, the Aramaic kolsa קולסא may be related to the Latin galea (plural galeas) - helmet.