If you open the context menu on your web project (e.g. by a right-click on the project in Solution Explorer), you will find the command Publish. You can use this command to publish your website in a number of ways.
One of them is to publish the website to a folder. In order not to overwrite any data, I recommend to backup the web.config and the html page that is present in your share up to now and then publish your website to this share.
To practice, you can first publish your website to a local folder and check the contents.
A good way to streamline this process is to either use the Release
configuration or create a specific configuration for the environment and use web.config transformations so that you avoid the need for manual changes to the web.config.
For a detailed overview on how to deploy web applications, see this link.