I think the solution depends heavily on your data access patterns.
I've used Azure Table Storage and it's great for many things. I've used DynamoDB and it's also good for quite a few things. Both are good table stores, but both have restrictions around read indexes, querying, and transactions. That's sometimes a show stopper. Both will require retooling your data and all the dependent applications.
For your file storage:
- (Cheapest, slowest) Migrate your files to a blob store (Azure Blob Storage or AWS S3) and leave them there. Use S3 as a drive for file access. This is slow, but cheap.
- (Performant) Use an EC2 instance with EBS drives and store your files there. Access the data on the local file system. This is durable and performant.
For your relational data, leave it relational and store it in a Cloud relational database server. (RDS+MySQL, RDS+SQL Server, SQL Azure, etc).
There's no need to change your applications, and their data patterns, moving to the cloud.