Most ASL teachers are indeed deaf and start right out using ASL, no voice. It's a good way to learn, IMO.
As far as classifiers go, this is something hearing people look at when learning sign. Not so much for the deaf, IMO. Not sure if that makes a difference in answering your question. I guess I find when deaf people are learning sign for the first time, they are picking up signs from other deaf people and/or from a sign language class. So classifiers aren't thought of as much in the deaf community. For example, the classifier 1 used to indicate a person walking, or the number one, etc., isn't thought of as the classifer 1, but simply the sign for the person walking or the number one. So I think all of us deaf people use classifiers without realizing it, but I'm not sure exactly how it is a learning tool for the hearies. Care to elaborate on why your teacher mentioned this, and the importance? I'm curious ...