Azure CDN is a 'pull' model & not a push model. You don't upload into the CDN, you would upload into Blob storage or onto a VM's storage in the /CDN virtual directory. On the initial request the CDN would grab the file from origin and populate the edge node. The second request would be served from the CDN.
The CDN does support video files & a variety of streaming codes.
Overview of CDN: https://azure.microsoft.com/en-us/documentation/articles/cdn-overview/
Best practices: http://blogs.msdn.com/b/windowsazure/archive/2011/03/18/best-practices-for-the-windows-azure-content-delivery-network.aspx
Another way to do this is to leverage the Azure Media Services: http://www.windowsazure.com/en-us/develop/Media-Services/how-to-guides/media-services-dotnet/#enable-cdn
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