You know it's important to sell yourself while interviewing. However I bet you're uncomfortable talking about yourself, your client. Tooting your own horn.
Aside from price, reputation, service, one of the most important ways you win that client is to stand out from your competition. To be more memorable. You must sell yourself. The best way to do so, and to get the points you'd like to make is to do it through a story. How you helped a different client, how your staff found a solution. How you worked well in a team. How you saved the day. How you saved a client money, time, embarrassment. Your accomplishments detailed via a story rather than an "I did this" will indeed make you more memorable and persuasive.
Telling a story well requires skill. Here are three tips to improve your success that I learnt from a speaking / sales coach, Patricia Fripp.
I/You Ratio
Portray the story entirely in the listeners world. Instead of beginning with "I cut costs by 10 per cent" change it to "I'm sure you have other clients that try to help you cut costs".
Description / Dialogue
Good stories have flesh and blood characters and dialogue. Tell your story through the conversations that occurred. Describe the people, the weather, set the scene. People think in pictures. Draw them with your words.