I'm currently advising a client right on exactly that kind of customer service integration.
Look into Live Ninja (https://liveninja.com/). It brings together the best of both worlds from web and a call center. Voice + Video + Instant connect without any special software, or a phone. You can have either one way, or two way video, and text chat too.
Obviously, get nice uniforms for your call center guys if you haven't already.
Simple, first understand the user need and their day to day activity/task and mental model. This will guarantee you best ROI.
Check following links theses will help you to understand best UX practices:
Before you expand into integrating a website with call centre keep in mind that there will be users sitting on the other end will judge you.
Users need to have an emotional connection to the experience of using your product. If you are merely creating an interface and not an experience, you have limited chance of gaining the following you need to make your product a success.
You can read more here: https://clarity.fm/questions/3298/what-are-the-best-practises-for-the-best-user-and-customer-experience-when
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath