At this time there is no way to perform an in place upgrade of virtual machines running one version of Windows to another. For example, upgrading from Windows Server 2012R2 to Windows Server 2016 from inside the Guest OS is not supported.
In order to upgrade the machine to a new version of Windows it is required that a new machine is created using the desired Operating System and the data migrated.
If an update is initialized from inside the OS the result will be a complete loss of RDP access.
In order to restore RDP access from that point console access is needed. Since console access is not possible in Azure the VHD will need to be downloaded on premises and spun up in an on premise virtual environment such as Hyper-V. Once the OS upgrade is completed the machine will need to be configured and uploaded back to Azure.
Please see KB2721672 for additional information regarding in-place upgrades inside of Azure Virtual Machines.
EDIT 9/6/17: As of today, you can now gain console access using Nested Virtualization. Please see the following articles on how to achieve this:
This post will be updated shortly to reflect all these changes.