Give them something to buy
The best way to land that first client is to give them something to buy. Ask for the order. Send a simple proposal and ask for money in exchange for your product or service.
Because what you are selling is new, you are selling yourself, so don't worry about the materials, website, business, card or case study. Entrepreneurs tend to complicate things (me too). When starting your business or rolling out a new product or service, remember that you are selling yourself or company first. Find a trusted business relationship and get feedback. They either believe you or they don't. Don't get too hung up on all the details. What you need are feedback and cash.