Creating a Visio Process Repository
Firstly, you need to have the correct level of subscription in Office365 to create a Visio Process Repository. This is a site template, so one way of creating a subsite is to open the Site Contents page from your SharePoint site, as in the following screenshot:
Then you can select to add a new subsite, by selecting the command at the bottom of the page (easy to miss if you don't scroll down) shown as follows:
When you click to add a new subsite, you get presented with a list of different site templates. You will find Visio Process Repository under the Enterprise tab under the Select a template label, as shown in the following screenshot:
Once you have created a Visio Process Repository, you will find that there is a Process Diagrams library that is already created for you. You can add more of these special libraries if you want to, by adding a Process Diagram Library app to the site contents. The next screenshot shows the new subsite before any process...