You can't run Varnish on a website which is actually not just standard IIS (see: http://www.slideshare.net/maartenba/windows-azure-web-sites-things-they-dont-teach-kids-in-school-comunity-day-2013). As others have said you'll need to run a VM to install Varnish.
how to install Varnish on windows azure?
Question
I just 'quick create' a website in windows azure with the simple step below and it's an IIS8.0 based site.
How to install Varnish in front of this IIS server in windows azure?
Solution 2
OTHER TIPS
I believe that you must change to Virtual Machine execution model to install varnish cache.
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