The Online Etymology Dictionary provides the following etymology:
c.1386, "soda ash," from M.L. alkali, from Arabic al-qaliy "the ashes" (of saltwort, a plant growing in alkaline soils), from qalay "to roast in a pan."The Arabic word is cognate with the Hebrew root קלה meaning "to roast, parch", and found in the Biblical words kali קלי - "parched corn" and kalui קלוי - "roasted, parched", and today a more official term for "toasted" (the slang is טוסט.)
Another word deriving from the same Arabic root is kalium, which is the earlier name for the element potassium, and is still used in German, Russian, and other European languages. This is why the symbol for the element is "K" - it goes back to the kuf in kalui (which Steinberg says is related to tzalui צלוי - also "roasted".)
By the way, the Sephardic surname Alcalay / Alkalai isn't connected to the above. It is spelled in Hebrew אלקלעי (with an ayin) and according to Stahl is related to the Arabic word kala קלעה - meaning "fortress".