You need to set up Webdeploy in the azure VM. it is same as an on-premises server.
Web Deploy Installation.. You need to install prerequisites - .net framework.
Configuring IIS 7 with webdeploy / Or with IIS8
Set up endpoints on azure vm to do webdeploy. Port 8172 for Webdeploy.
Then you can create a publish profile in Visual Studio 2012/13 to deploy to this endpoint. http://msdn.microsoft.com/en-us/library/dd465337(v=vs.110).aspx
or You can do it using Powershell scripts. sample- http://gallery.technet.microsoft.com/scriptcenter/Deploy-a-Web-App-to-Two-99f1a3bc