Microsoft azure cloud provides SMB 2.1 file server functionality.
This is great when you need to access your files from different VM or services,
since don't need to run your own VM with the fileserver role installed.
As a bonus, you can access your files also via REST service, so your are very flexible in the usage of the service.
A introduction to the fileserver functionality can be found in this blog post.
When you have setup your service correctly, it's time to move some data to it.
The simplest approach would be to use xcopy or a similar tool to upload your files into the azure file service.
The drawback of such standard tools is, that they copy the files sequencially, one by one.
Azure is a cloud which provides massive paralell operations, so there is a faster way to get your file into the cloud.
Microsoft has released a tool named AzCopy which allows you to upload whole folders in paralell, which will speedup your uploads.
In this blog post there are more examples and use cases to bring your files into the Azure cloud.