We have a number of one-click deployments running in TeamCity. It's honestly very straightforward. If you're deploying to Azure you can use either the Powershell or node.js libraries to talk to their API. We use powershell. Basically, we compile using msbuild, package the artifacts into an Azure package, and deploy it via the API. Doing zero-downtime deploys to Azure is a matter of how you use the API. You can substitute EC2, another cloud, or traditional hosting and while the tools and APIs get progressively less powerful, it's all doable. At the simplest level you may just use msdeploy as your "API."

I'm a huge believer in this kind of automation. We make extensive use of TeamCity to drive it. In addition to builds, automated tests, and deployments, we also use it for release promotion and doing other housekeeping tasks like cleaning up data from QA environments, restoring testing databases, etc.