Before you begin you need to understand how Microsoft team has designed their Architecture and how it reals works, to know this review my earlier blog post referencing the Share point conference video session "What's behind Office 365?" once you review this video, you will have a strong idea over how things are setup and work in the backend.
Now its time for us to review the TechNet blog post, when you read this article ensure that you capture the differences between how each client connects to different Office 365 service and how it varies from one another, for Example: SharePoint Online Client connects to the Actual datacenter and Lync Online Client connects to the nearest Regional datacenter etc. Having clear idea in this area will make you understand how the actual connection is established when your clients tries to access the service internally within Office 365 datacenters and provides you adequate knowledge to troubleshoot issues if the client is not able to make successful connection with the service.
Review the TechNet Article here: Reference: Client connectivity
Ensure that your Organization meets the below critical requirements for the clients to connect to the Office 365 service, These critical requirements include System requirements, Network Connectivity Configuration and DNS records, there are other requirements too and these areas are critical and most of the issues occur in this part if its not setup as desired, so as stated earlier ensure that your Environment is setup properly in-accordance with the guideline provided by Microsoft.
Office 365 system requirements
External DNS records for Office 365
Office 365 URLs and IP address ranges
If your Organization is fine with these above key areas then you will likely have a highly available solution with the Office 365 service.