Preety much knux, case studies are the way to go. The majority of designers only concentrate on designing, they don't even think about the business implications that their design brings. Businesses don't buy a design because their old one isn't nice, they buy a new design because it isn't doing the job e.g Converting enough sales, providing clear and concise information.

As a designer you are required to provide a solution to their problem, a case study as moobrack said is an in depth study/treatment of a particular website where you outline the problems you face, the clients specification etc. You would then review your progress/outcome/conclusion after the design is finished and provide feedback and perhaps some statistical information from the client to back it up e.g Sales went up 15% since we implemented our new design.

Check out www.ceonex.com for a better idea of case studies, they do this very well.