I have the 8x8 asbestos floor tile in my basement.
I want to install new vinyl tile.
What is the best way to do this?

My thought is to put down a thin layer of durrock over the existing tile and then installing the new vinyl on top of that.

How should I secure the durrock to the old vinyl tile?

If there are no loose tiles, clean them, put down a flooring primer and peel and stick over the asbestos. No need to install a subfloor unless needed.