I agree with Ed that the main problem is the word 'run'. 'Hosts' is a good replacement, or 'presents'. So, with minimal changes to your original sentence, and keeping it in (correctly spelt) British English, you could say: He's a famous presenter who hosts TV programmes featuring animals.

In Us English, we don't call them "presenters". As Jerome said, we'd use "personality" or "host". And we usually use "show" rather than "program". I think your terms are fine in UK English. However, I'd write this a little differently because "run" isn't quite the right word. Also, by "devoted" do you mean the animals are performers or are shown in nature? He is famous for hosting TV shows featuring animal acts. He is famous for hosting TV shows featuring animals in their native habitats.

He is a famous TV personality that runs programs devoted to animals. Presenters on TV are still considered TV personalities. Presenting is just one of the many things they usually do.