a voltage meter does not read voltage, it reads potential difference. If you want to check the voltage you need to connect the positive lead of the meter to the wire you wish to test and the negative lead to ground, not the other wire.
If the light is on and you put the leads of the meter on the two wires for the bulb, the meter will only read the voltage drop through the bulb.
The positive wire for the light is a good source of power to connect to the "control" pin of a relay. Install an inline fuse for this wire, a 2.5A fuse should be plenty. This way if there is a problem with the relay or the wire it won't knock out the whole circut you tapped into.
For the "power in" pin of the relay I suggest running a wire from a fuse on the fuse panel that has power all the time even when the ignition is off. Or you can connect directly to the battery. Always be sure to install an inline fuse in the wire.
For the "power out" pin of the relay run that wire to the lights you wish to run
and the ground pin of the relay well that should explain itself.
