Since Microsoft released Windows Azure Virtual Machines feature in June 2012 we have an ability to run Linux in Windows Azure VM’s. We have partnered with a few Linux distribution owners to provide supported version of Linux. These Linux distributions are available as a part of the image library.
List of endorsed distributions is shown here:
You can find all the documentation about running Linux on Windows Azure VM here:
You can also bring your own Linux image to Windows Azure. To do this you will need to run Linux in Hyper-V and Install Windows Azure Linux Agent. After this you can use csupload tool provided with Windows Azure SDK to upload this image to Windows Azure.
You can download the ISO file
containing the agent here : http://www.linuxonhyperv.com/waagent/WALinuxAgent.iso
The source code of the agent can be downloaded from here : https://github.com/Windows-Azure/WALinuxAgent/