To deploy a simple site using Postman and the Netlify API, you need a personal access token, a simple site to experiment with, and Postman. You log in to your Netlify account, get a personal access token, clone the repository, and review its contents. Then, you create a POST request to create a new site, followed by a POST request to create a deploy with the list of files that will be included. Next, you make PUT requests to upload each file individually, using the deploy ID from the previous step. Once all files have been received by Netlify, your deploy is complete, and the site dashboard shows that it has been deployed successfully. This process provides more control over the deploy process and can be used for building integrations or automating workflows that connect with other systems.