There are basically two ways, to make the coat follow and deform with the armature.
The easiest way is to parent the coat to the armature with so called automatic weights. To do that
- select the armature and delete the parent in its object tab (this is perhaps a remain from a previous attempt
- then select the coat and shift select the armature
- after that hit CRTL P and select “with automatic weights”
After this procedure it will transform when you pose the armature. You can adjust the weight painting of the coat to your needs. If you add collar bones to the armature it will help at the shoulders, because these will peek through the coat at the moment. If there are still parts of the body peeking through after all that, you could have a look at the mask modifier, which helps to hide body parts that are inside the clothing.
As yogyog said, you could also use the cloth modifier for the coat. That’s more accurate because it calculates how the cloth would react. To do this add a cloth modifier to the coat and a collision modifier to the body. Then experiment and adjust the settings of each modifier until you are satisfied with the result. But be aware that this adds calculation time to the animation. An even more after you may add a subdivision modifier to the coat.
With a combination of
- armature parenting
- cloth modifier
- collision modifier and
- subdivison modifier
you will get this (no fine tuning for cloth and collision):
Hope that helps