How is the actual ground speed of an airliner measured?
There are generally two ways of determining the ground speed from within the aircraft.

The first is by using an inertial navigation system. This consists of a number of accelerometers and gyroscopes that measure all accelerations and rotations of the aircraft throughout a flight. By mathematically integrating all measurements this navigation system is able to compute the speed and position of the aircraft at any time. The accuracy will degrade over time but ground speed is usually not far off, otherwise position would be far off very soon. 

The second method uses external radio signals. Nowadays this is mostly GPS, but ground speed is also determined by using a number of VOR (VHF Omnidirection Range) and/or DME (Distance Measuring Equipment) stations. These navigation beacons are widely available over most continents. A navigation computer would select the right set of beacons and by analysing the rate of change of the beacon bearings / distances the aircraft's speed can be determined. 

Modern navigation systems combine all these methods into a one solution. GPS, inertial navigation and DME measurements are then fused mathematically together to obtain a highly accurate and robust position and ground speed.