The correct form is choosing. The word comes from the verb “choose,” which changes by adding -ing to become “choosing.” The spelling “chosing” is incorrect and commonly mistaken because it seems to follow the pattern of dropping letters found in other English words like “losing” (from “lose”). For example, when deciding what to eat, you might say, “I am choosing between ...
Choose is the most general of these words, but always represents an act of the will; it is the taking of one or some where all are not wanted or cannot be had. Choice may be founded upon preference or modified by necessity.