We use Scrum and other agile methods to deploy complex software frequently and incrementally in order to get frequent feedback on the implemented features of the product.
Exactly the same applies to implementing process changes. We want to avoid big-bang installations within software development as well as within process changes, and we most certainly want to get frequent feedback on our process improvements.
Then why not use Scrum to implement Scrum within an organisation?
You might want to start in a small scale with one chosen project, and then scale up to include more projects step by step. A product owner and a scrum master is appointed, which may vary from organisation to organisation. Also a backlog should be established early on.
With the roles distributed and a backlog established the rest of the process can be implemented piece by piece and added to the backlog: people should be educated and coached, which project to go next, retrospectives must start up, timeboxing should be adopted if not in place already, performance measurements should be started, teams should be formed, other agile practices like continuous integration could be evaluated and adopted etc.
All this cannot be done in one big step and should not be done without the help of experienced scrum/agile coaches. Even on project level scrum should be adopted incrementally and process improvements should continuously be added to the product backlog of the project and being evaluated in the retrospectives.
Implementing Scrum is about changing the mind-set of people, and that is not done in a twinkling.
I have been working as a software consultant for more than 11 years. Because of that I am an eager supporter of lean principles and agile methods.