I have started several PMO departments for companies that have demonstrated a need to have one.
The objective of a PMO department is to centralize how you are delivering projects. It usually happens when you have too many projects, or ill equipped project managers and there is misalignment between your projects and your company's objectives.
The nature of a project is to create a change within the organization. When you have too many projects, you have too many competing changes in progress.
There is not a single strategy on building a PMO department because the business rationale for one will vary. For this reason, starting a PMO department should be treated as its own project. Define a business plan, change management plan, budget, and ensure you have stakeholder support. Only until you have done the proper research and planning should you embark on building a PMO department.
PMO departments will carry serious overhead and your organization needs to weigh the pros/cons of this based on the changes your organization is trying to undertake using a project management approach.