From my experience and reading, Files Connect and External Objects are read-only and thus this must be accomplished using through API calls to SharePoint.

We created a product in the App Exchange called Trove which allows users to upload documents into SharePoint from Salesforce and associate those to Accounts and Opportunities, but as of today, Trove doesn't utilize Files Connect.  

For a solution which utilizes Files Connect, I envision the solution for uploading files would be custom Apex/Visualforce page.  I would use the SharePoint REST API for Files (https://msdn.microsoft.com/en-us/library/office/dn450841.aspx).  You'll want to be sure to use the "chunked" method for uploading files which involves using BeginUpload/ContinueUpload/FinishUpload API methods of the SharePoint REST API so that you don't hit the 6MB heap size limit in Apex code.  Also, you'll need to decide where to store files in SharePoint:  Does the user choose the library and folder on their own or will the custom Apex code choose the library/folder for them.

Authentication into SharePoint Online is a bit of a challenge, but it is made easier by using Auth Providers and Named Credentials that are built into Salesforce.  You can also code your own OAuth flow if you have custom needs.

I imagine I provided more answer than you're after right now but hopefully that was helpful.  Today, I happen to be exploring Files Connect myself.  I'm happy to share more details and experience if it's helpful.