i think the best method to convert video files using ffmpeg is : making a web method (Wcf or WebApi) in a web server if you have a server. You can create a simply file upload method and after , u have to call your ffmpeg code on your server and finally u have to download the converted video to phone.
Or you can create an Azure account and create a method (wcf or api) and host it in Azure. With blob storage you can keep files in Azure blob storage.In this scenario you'll host your videos on Azure Blobs.Upload and download files is very easy with AzureSdk and apis. And the best part of Azure Blobs you can host ffmpeg in blob and you can run the exe(ffmpeg) in WebRole or WorkerRole.
You can check this examples here:
What is Azure Blob
Running exe in a WebRole