Smooth Streaming Video from Blob Storage
I have been doing almost exactly this, but without the Adaptive Streaming (I'm dealing with video files of only 25mb or smaller). I am using JWPlayer for the front end in my web page, and the video is in Azure blob storage.
I'm using the pre-built windows binary ffmpeg.exe I uploaded it to Azure (put it in root of Worker Role project), then I use System.Diagnostics.Process in a WorkerRole to massage the video files. You could do the same to create the different quality versions you want.
So, when a user uploads a video my business logic layer plops a message in an Azure message queue, my worker role picks it up, copies the video onto Azure LocalStorage (LocalStorage is physical disk you can configure in your Azure project settings file), runs ffmpeg.exe over the file and then puts it back in blob storage.