Al-oh-wish-us is the pronunciation most often used in English, and the one that I have heard the most, since it is a name used by many of my "Great" relatives and the deacon at my parish (we call him Deacon Al.) The other pronunciation A-loy-see-us, as seen in the description of the name on this site, is a Dutch pronunciation.
Aloysius is my grandfather's middle name and he's always pronounced it al-o-wish-us. I don't know if that's an approved alterantive pronunciation. He may pronounce it that way because of his Irish accent.