SQL2012 - Send backup to client via SQL Job
Assuming that all the various security concerns are met; I would use SQL Agent to kick off a PowerShell Step that would move the file for you.  Depending on where you end up pushing the file to this can be easier or harder.  

Depending on what version of SQL you are using, you also have the option of backing up directly to Azure Blob storage, builtin. More details here Backup and Restore with Azure Blob Storage