If I have a clean plate and then another layer with talent acting in the same exact shot as the clean plate, is there any way for After Effects to remove the background of the shot with the talent? I essentially want to do a green screen without the green screen.
You need to use a technique called rotoscoping to separate the subject from the background. You can then put the subject over the clean plate.
Rotoscoping is not an easy or an automatic technique. You'll find valuable information about it here.
This page has some very good links. You should know that AE's Rotobursh tool can not be used intuitively, and that you must look at the training for it that you'll find in the links in the page above.
You can try Mylenium's Difference Matte suggestion. It might work, depending on the nature of your footage.
But keep a sharp eye on the edges of your subject: if they don't look good to you, they probably will never look great using it. There aren't many ways to tweak a Difference Matte.
If there were easy and accurate techniques to avoid shooting chroma keys, the whole planet would be using them, and chroma keys would be dead. They aren't.