CPU usage goes up while computer enters into screensaver/screen goes dark

Your computer is possibly running Scheduled Tasks that are set work when the system is idle, for example, defragging in the background.

Note the "Start this task only if the computer is idle for" parameter in the screenshot.

In the Task Scheduler, I would also try clicking on the "Display all running tasks" in the right pane. I am not sure if one of these idle tasks will show there, but it might.