Copy files to Azure VM
In my case simple restart of rdpclip.exe
did the job, so try it.
You should just be able to copy/paste like normal. You can check to make sure rdpclip.exe is running on the VM.
If that doesn't work, you can always open your local drive using \\tsclient\c from within the RDP session. To share your local drive you can save the RDP file from the Management Portal website, then right-click the .rdp file and select Edit. Then switch to the Local Resources tab and click More under 'Local devices and resources' and check the drives that you want to share.