How do Chinese people say it when they are proud of someone? This is not a question about the literal translation of the sentence: "I'm proud that you are my dad." The question itself is more like: what do Chinese people say when they want to express their feelings to someone that they are proud of? Like, when children are proud of their dad?
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."