Retrofit cnc 3axis

You need to connect one lead of the voltmeter to GROUND and the other lead to your signal.
So to start out connect the voltmeter leads to pin13 and pin25 of the db25 connector you should read 24V. If you read -24V then switch the leads around.
In order to know what voltage your motor drives expect you need to connect the pc and enable the machine then you can measure the voltage of pin 5 AX_ON on db25 (ie one test lead to Ground (pin13) the other lead to pin 5)
To measure the voltage of the DIR pins move the lead from pin5 to pin14 DIRX and jog the X axis first one way and then the other way. You should see a voltage change when you change the direction.
To measure resistance or continuity anywhere you want the pc / controller to be switched off.

Could it be that the signals are at 5v?

Yes, as a matter of fact that is what I expected to come from the PC but the "24V" in the pinout of db25 on the schematics made me wonder if maybe the PC card is switching 24V coming from the controller.

If I have a 3.8V result does it mean that the ground is in that pin anyway? 

Not necessarily. What you could try is to use the ohmmeter to see if a pin on the db25 connector is connected to the case of the controller as the Ground might well be connected to the body of the controller, the pc and or the machine. Remember to only use the ohmmeter with the controller/pc switched off.

to do the other tests with the axes enabled, do I have to open the cable socket and test the pins from behind? 

Yes, you need to find a way to probe with everything connected.

Do I connect one end by testing each of the DB25 pins and put the other in contact with the iron of the machine cabinet or the machine itself?

Yes, if you find a pin with continuity make sure to switch the meters leads around and check that pin again.

Good, now with that result you can permanently attach the meters ground lead to the case or the ground pins and probe the other pins with the other lead.
What you want to look for is changes in voltage on the individual pins. ie the pin to enable the motor drives may change from ~5V to ~0V (or from ~0V to ~5V) when you enable the motors.
Same for the DIR pins when you change the direction of an axis. Same again for the toolchanger functions. The STEP pins will show a voltage somewhere between 0..5V during jogging of the axis as the meter will be reading an average of a rectangular signal.

edit:
You want to note down if the change is from 0 to 5 or the other way around to activate a function as that will be important later.
function active when pin is 0V -> active low
function active when pin is 5V -> active high

0.8mm is fine.