First, you'll have to set the widget to not appear ( Settings > Widget > Settings tab > check the box next to Turn off chat widget in the Hide Widget section). You cannot set the widget to not appear if you are using a Lite plan.
Now, you need to do is use the API below along with your custom image/button and it'll pop open the chat window when it's clicked on:
The entire script should look a little something like this:
Make sure to update "INSERT_IMG_SRC_HERE" on the code above to the source of your image, like "domain.com/livechat.png".
With that, you should have your very own button/image that'll open up the chat window when clicked on.
Note that you'll still need to have the actual Chat script on your website's coding for this to work properly.