How do Chinese people say it when they are proud of someone?
The simplest sentence would be something like: 
我为[insert person]骄傲。
So in your case：我为爸爸骄傲。
Slightly more sophisticated: 
我为你感到自豪。
@Ha Hae's 我以你为荣 would also work but would translate to something like “You are in my place of honour."