Normally just one coat of primer. If a little of the old color is showing thru you will probably be o.k. and the top coat should cover.
Not a big fan of Kilz...
Sounds like you are already in progress with the paint.
On future projects I would suggest going to the big boy paint store versus the big box. They will stock a lot better grade of primer and can tint it a shade or two lighter than your top coat so you'll get better end color with just one coat. Especially on major color changes.
Yes, you'll pay a few more bucks a gallon for the paint, but most painting is the prep work not the material cost.