You can't do it quite the way you describe. You need to learn how to edit a wordpress template/design and adjust to the way WP does things.
Few people use Illustrator for building web graphics but I do. But don't use Illustrator slices. Don't let any graphics program write code for you. Instead use the Illustrator artboard tool to define the graphic chunks.
Look at Adobe Business Catalyst which includes InContext CMS with your hosting.
Check out Konductor CMS if you want full design control and no server-side setup. The interface for your clients is fully in-context as well.