I have never seen any of his father's cars
This is correct. This is the one that any native speaker would use.
I have never seen any cars of his father
This is wrong, however "I have never seen any of the cars of his father" is, as far as I am aware, grammatically correct. But please never write this, it sounds incredibly awkward and no native speaker would speak or write this way unless they were intentionally trying to sound like a Norse god or something.
I have never seen his father's any cars.
This is completely wrong, I couldn't even understand it without seeing the other two sentences.
In general, we use an apostrophe for possession. There are, however, times when a preposition is used (particularly when trying to sound important). Things like titles (The Lord of the Rings rather than The Rings' Lord) or announcements (Will the owner of the blue car outside please move their vehicle) are places that you might use a preposition.