If you hadn't painted over it with the KILZ, then you could just mix your thin set powder with straight additive (pronounced "adhesive"), and tile right over that black asphaltic adhesive.
Now that there's a coat of KILZ over it, you should check to see if the KILZ is sticking well. Try scraping or peeling it off, and see if it puts up a fight to stay on. If it does, then it's sticking well to the black adhesive, and I'd leave it in place.
What you need to do is trowel down a bed of thin set and lay the cement board in it, and screw the cement board down. This layer of thin set will fill in any gaps between the existing floor and the cement board that would flex if there were no thin set between them.
For floor tiles, rather than use a cement board like Wonderboard or Durock, you're better off using a product called "Hardiebacker board" that comes in 1/4, and 1/2 inch thicknesses. Hardibacker is whitish in colour and is much stronger than the traditional grey cement based boards like Wonderboard.
I'd find out if Hardiebacker comes in 3/8 inch. I've never done any floor tiling, but I've set way more than my fair share of wall tiles.
Then you set your ceramic tile over the Hardiebacker board.