Is CC automatically deactivated on the old PC when it is installed on the new PC.
No. Activation/deactivation is controlled by the user. With each subscription, you have two activations to manage as you wish between devices.
For example, you could install CC on 5 devices and juggle 2 activations (Help > Sign In/Sign Out) between them as you use CC on different devices. In that scenario, at least 3 devices would remain deactivated at any one time.
To deactivate on the old PC, open any Cloud app, e.g. Photoshop and click Help > Sign Out. That will deactivate all Cloud apps on that PC and free up that seat for use on another PC or Mac.