Ginger Daniel has started a series on data migration into Azure Blob Storage:
Part 1 of this article demonstrates how to upload multiple tables from an on-premise SQL Server to an Azure Blob Storage account as csv files. I covered these basic steps to get data from one place to the other using Azure Data Factory, however there are many other alternative ways to accomplish this, and many details in these steps that were not covered. For a deep-dive into the details you can start here https://docs.microsoft.com/en-us/azure/data-factory/introduction, and https://docs.microsoft.com/en-us/azure/data-factory/quickstart-create-data-factory-portal#create-a-pipeline.
Part 1 was chock full of information, and it looks like Part 2 will be as well.