Neither a lock screen nor a screen saver will prevent your backup program - or any other program - from continuing to run. But if you set your power options to sleep, hibernate or power off after a specified length of time, that will stop all programs from running.
It's certainly confusing, so I'll explain:
A lock screen and a screen saver are overlays - images that are placed 'on top' of your desktop, but they are used for different reasons. A lockscreen image is intended to prevent passers-by from seeing what's on your desktop - it's a privacy thing. The reason it's called a lock screen is because it's normally password-protected, so the overlaid image is 'locked' in place until you enter the password. Your computer continues to run normally with a lockscreen.
A screen saver is a moving image intended to prevent burn-in on old CRT monitors. You can password-protect a screen saver also, if you want to. Like with a lock screen, your computer continues to run normally with a screen saver.
The power options (Control Panel > Power Options) are different. Their purpose is to reduce your computer's use of electricity. You need to be careful here, because many power options are setup by your computer manufacturer at their factory, so it's worthwhile to go over these settings yourself.
One way that your computer can save electricity is to power off the monitor after X minutes. Even when the monitor is powered off, your computer is still working, and all your programs are still running. You just can't see them.
However, if you set the power options to power off your hard drive, or to 'sleep' the computer, or to hibernate the computer, that will bring all processing activity to a close, and your programs will stop running.
(To see the full extent of the power options, you'll need to be signed in to an Administrator account.)