You remove the bolts and 'rotate' the alternator counterclockwise (more or less) and work it out of the arms holding it.
You then kinda spin it over such that the pulley is facing downward and remove the plug and cable from behind and THEN work it around and get it out by moving it directly forward towards the radiator.

I don't know if you can do this from below, but I cant, as I have not lift. I"ve replaced my alternator four times in five years (DO NOT get an Alternator from NAPA, their rebuilds are awful).