What details prove that Charlie's intelligence is growing and increasing daily?

One detail can be found in the opening paragraph. Charlie mentions how Dr. Strauss is angry at him for not having written a progress report in two weeks. Charlie understands why the doctor's mad at him; after all, the lab's now paying him a regular salary and, not unreasonably, expects a return on its investment. In his defense, Charlie says that he was couldn't write a report because he was too busy thinking and reading. Too busy thinking and reading? Let that just sink in for a minute. Can you imagine the Charlie of old ever saying something like this? Absolutely not. But because his learning's coming along in leaps and bounds, we can well believe that this is how Charlie now spends most of his free time.
Charlie's growing intelligence now also means that he can feel free to evaluate the intellectual credentials of scientists working at the institute. For example, he loftily declares that Dr. Nemur is no genius; yes, he has a good mind, but struggles under the specter of self-doubt. As for Dr. Strauss, well, he is a genius in Charlie's not very humble opinion, but his areas of knowledge are too limited. The only ancient languages he can read are Latin, Greek, and Hebrew—that's three more ancient languages than most of us can read, but according to the super-smart Charlie, it's a sign of serious intellectual deficiency on Dr. Strauss's part.