Earlier this month I released a Hugo Docker image right in our forum that can be used to build and test your Hugo website. I suggest trying it out, I use it for several websites already (feedback welcome).
Now I’ve written a blog post on how to use that Docker image (or others) to build, test, and then deploy your site all within CircleCI. You can read this new blog post here. It covers the config for 2.0, how to use the Docker image, etc.
My blog post covers the whole process from start to end, using CircleCI 2.0, which is Docker based (and very fast). For some reason, if you only want to use CircleCI 1.0, a Hugo forum user @dundonian has made a great post on how to do that here.