Alchemy Points are necessary for using spells in battle. Intelligence points are gained from reading books, or playing Mystic Spindles and raising the intelligence stat, or by using stat boosters, or randomly by zapping. 1 alchemy point is obtained on the 16th multiple of intelligence, but only if you read a book for that point. If you forget, or if a stat booster pushes you over this point, you can regain the lost point on your next multiple of 16 if you make sure to read a book.
There is an Alchemist in the Underground in the Shadowglen. Alchemy Points are irrelevant to this.