Q. Is it AKA, aka, or a.k.a.? What about in dialogue?
A. Chicago prefers “a.k.a.” (with periods; see CMOS 10.4). As for dialogue, the answer isn’t “A kay A”—nor is it “also known as.” Writers are sometimes under the impression that because dialogue is supposed to represent speech, abbreviations (and numbers) should always be spelled or sounded out. But if the pronunciation of the abbreviation would be understood by most readers, you can write it as you normally would:
“I’m going to the store, a.k.a. the shop,” my roommate informed me. “Would you like me to get you anything?”
The same goes for an abbreviation like “Ms.” or “NASA” that would be understood (and said) as a word rather than as a series of letters. For some additional thoughts on abbreviations in dialogue, see this post at CMOS Shop Talk.