At the time of writing, the Azure Application Proxy makes it easy to pubish an on-prem web application to Azure. There are tons of resources explaining how to do that and it is a straightforward process.
Go register to the COLLAB365 Global Conference, a 24hours online conference on SharePoint, Office 365 and Azure
As many others, I'll be speaking at a very promising online event with over 100 speakers and many many sessions on Office 365, SharePoint, Azure and more. Go register (for free) on the official site http://www.collab365.events/
At the time of writing (08/2015), Office 365 is really becoming a great platform with a lot of APIs for a very affordable and competitive price. As I'm currently involved in OneDrive for Business implementation, I focused on the File API whose the main endpoints is:
and I consider it a good start. The integration with Azure Active Directory is awesome (also with the other services) but the API itself lacks two major features to rock: