When we are getting the measurements of speeds of rockets. Is this calculation done through external systems watching the rocket or can the rocket measure its own speed relative to the ground?
You can put measuring devices anywhere you like, but the best answers are always computed by measuring many different ways simultaneously. If you have enough independent methods, you can detect and correct some common kinds of errors. Measurements are only half of the story; you also need a good navigation algorithm to take a bunch of different kinds of measurements and turn them into a best estimate of position and velocity. This is generally done with some variant of a Kalman filter, since that's what navigation engineers call the most basic way of using measurements to iteratively update where your model thinks you ought to be.
An inertial measurement unit (IMU) is extremely common, but better ones are bigger and more expensive. The core problem with any IMU is that it measures acceleration, which needs to be integrated to compute velocity, and then integrated again to compute position. Therefore, any random bias in the IMU due to misalignment, inaccurate calibration, or other error that is consistent during flight causes the inertial navigation system (INS)'s computation of velocity to drift away from true roughly linearly with time, while position drifts quadratically. The way to fix that is have some completely different measurement method, like star tracking cameras or a GPS receiver, to give an independent check on where the rocket actually is, which you can compare to your model of where it ought to be if the measurements were right, and then calculate a correction to the numbers the IMU is giving you.
Radar is nice, but it only measures range, and possibly range rate, because only motion along the line of sight causes a Doppler shift, and you have to point it at something that bounces it back (down at the ground, not up into the sky). A rearward-looking radar onboard the rocket could give you pretty good answers for the first few seconds, while you are going straight up rather slowly; but once you start to pitch over, things get more complicated. If you have a good enough map of the region you are traveling over, and you're flying fairly level and not too high, then a radar altimeter can be used for terrain contour matching (TERCOM), a process long used for cruise missile navigation; but it doesn't apply to space launches, because the trajectories are so different. The important radars in rocket motion determination near takeoff are the range instrumentation radars scattered around the launch site, which can combine observations of the same rocket from several different well-surveyed locations. See, for example, Nessmith 1976 (IEEE Trans Aero Elec Sys 12.6, 756-766) for the early history of these devices. Those observations, together with telemetry from the rocket (including the IMU and any other available sensors, as well as the rocket's navigation filter's best estimate of its own position), are generally fused by a different filter on the ground to provide an improved answer.
I don't know for sure what any particular launch vendor uses, but they don't really need to tell the rocket to update its onboard state estimate, if they are instead directly commanding the thrusters; there will be time to update everything during the pauses between later burns, again as part of determining what to command the rocket to do. I don't know for sure which gets displayed at the bottom of the video screen; but if it were me, I would publish only the ground control station's filter solution, not any of the raw telemetry.