Both cloudlinux and openvz are user space virtualization technologies, so I would never use them together or one inside the other, as they may conflict with each other.
I did however have a great setup which had XEN PVonHVM as main virtualization and CloudLinux as user space virtualization. Meaning that we installed CloudLinux on a XEN VPS, and it worked great in production for years.
CloudLinux mainly focuses on the providing the best user interface and provide the smooth access for the managing the server resources, inodes. It has the LVE manager that is LightWeight Virtual Environment.
In CloudLinux hosting, the web hosting account which is consuming the more resources and causing the server to run slower will be automatically stop working without affecting the other sites on the server. This was not previously possible in the traditional hosting server.
You can assign the particular set of resources for the individual account on the cloud linux. You will have multiple versions of PHP which you can change just by one click.
There are limited support available with OpenVZ on CloudLinux.
If your on OpenVZ then you shouldn't have to even worry about Cloudlinux. But rather upgrade to at least a quality KVM/Xen solution or a dedicated box later on. Then add Cloudlinux then.
Even a KVM provider said because of the costs of the licensing that they advise just making sure you have enough available resources then transition to Cloudlinux once you have enough accounts to make licensing worth it.