That's a good question. It's not just education but also experience. A background in computer science is, of course, recommended. Naturally, understanding hypervisors (such as VMware, Hyper-V and OpenStack) and service management concepts (such as those provided by ITIL) are huge contributors to becoming good at designing just about any cloud-type platform. On top of this, it is important to develop competencies in technology, data, application and information architecture disciplines (which methodologies like TOGAF, Macroscope and Zachman can help to provide).
To give an example from my experience, I achieved a BA (Hons) in Computer Science in 2007 and thereafter spent my first 4 years in the capacity of a Systems Engineer. I furthered my education with an MSc in Management Information Systems which gave me a thorough appreciation of managing IT economics and service-based infrastructure from a business standpoint. My last 5 years have been spent doing solutions and enterprise architecture and during that time I leveraged and refined all my engineering knowledge and education into the art of cloud design. It was a fairly long road to travel, but completely worth it!
Thanks for your query, and have a great day.