I want to build a radar alert system with Arduino
I think that's quite possible. As mentioned, not technically radar. I don't know if cats could hear it, but then if it scares them off, job done, right?

I don't know if sending the alert to a PC would help or not - you might just consider an audible alarm using a cheap 8-ohm speaker. Otherwise, USB, bluetooth, wifi, or SMS for notification. A PC or phone (on all the time) to receive the data. Some thought about what you will do with it.

As a deterrent, a water pistol is ideal, you'd need a motor - solenoid? - to fire the water pistol. Harmless, but very distasteful to cats.

https://www.youtube.com/watch?v=lxmRYRlgreo

Update: Inspired by @SDSolar, I checked it out - yes, cats can hear that frequency. 


  Cats have excellent hearing and can detect an extremely broad range of frequencies. They can hear higher-pitched sounds than humans or most dogs, detecting frequencies from 55 Hz up to 79 kHz.


(from https://en.wikipedia.org/wiki/Hearing_range#Cats )

This means they will probably learn to associate the sound with the water pistol (yes, cats CAN be trained, if they are motivated). This means if you remove your device, they will probably come back, but if you leave the sensor (active), but remove the water, then it will continue to work!