From your BigCommerce dashboard, go to Advanced Settings ▸ Live Chat and check the Other Third Party Live Chat Service box. You'll need to click save in the bottom right before continuing.
Navigate to the Other Third Party Live Chat Service tab, paste your Pure Chat code in the text box, select At the top of the page from the dropdown, and hit save.
In the page editor, click on the Settings tab (top-center). Open your Pure Chat dashboard, click the toggle next to Unavailable on Chat Boxes, and refresh your web page to see Pure Chat on your site! Scroll down to the Advanced Options area and click Add Footer HTML.
Now your BigCommerce site will have your chat box!