How is the FPA (flight-path-angle) computed?
The flight path is a path defined with respect to the ground. Therefore, the flight path angle must be computed with the ground speed:



The flight path angle $\alpha$ is calculated by the on-board computers with:

$$ \alpha = \text{arctan} \left( \frac{V/S}{\text{GS}} \right) $$

The source of the ground speed would in general be aircraft dependent, but for an airliner this would come from the Air Data Inertial Reference System (ADIRS). See also How is the actual ground speed of an airliner measured?

Some non-precision approach charts will also have a table with required vertical speeds for various ground speeds to maintain the target flight path angle. This is particularly helpful if your aircraft cannot fly a given FPA itself. This is an example from Amsterdam Schiphol (EHAM) VOR Approach for runway 09:



If you e.g. fly with a ground speed of 150kt on this approach, you would just dial in 800ft/min and you would get the desired 3° FPA.