To examine my chin usually I just look in the mirror... (jk!)
You probably won't be able to examine his teeth. Usually a vet will put a chinchilla under anesthesia for that. Even my very tame chinchilla, Kulu, will not the the vet pry his mouth open to look in. (Some chins may allow for that but it's rare). A proper tooth exam requires special equipment. If your pet is eating and pooping normally and is at a healthy weight then there is no need for that. Sometimes if your chin does the grooming thing when you scratch his chest area you can get a look at the front incisors. I've done that with my last chin that had serious problems with them and didn't notice anything wrong other than a slight curvature. The vet found the teeth had overgrown and were cutting into the mouth. Weight loss and anorexia prompted me to get the exam done and it was not cheap. Basically looking at the teeth directly didn't help me much because in this case there were no serious abnormalities above the lip line.
You probably should wait until he is more tame before you try to restrain him to examine his feet yourself but if you get another person to help you so it only takes a few seconds you can try that. One thing you might be able to do if you have a glass tabletop or sheet of plexiglass is to put him on it and look up if you're concerned there is something there. I've used that method to examine our parrot's feet when he was having issues with sores and it works well. Just make sure he can't fall and hurt himself.
