The meter should have an A scale (Ampere) for reading current. Preferably, a High amp one (10A for example). In some of them, this high amp scale requires the red plug of the wire probes to be connected in a different hole. We do not know the magnitude of the current involved, so in order to preserve the meter, it is better to start reading in this high amp scale.
Disconnect the ground (negative) lead to the battery (electronic devices in the car will loose status when battery is unplugged, so you will need radio code for example), connect the positive (Red) meter probe to the negative thick wire disconnected from the battery, and connect the negative (Black) probe of the meter to the negative battery post. Be sure that the probes are inserted in the proper holes in the meter for the readings (Amp) you intend to take.
If you have the leak as you stated before, you should have a significant reading in the meter (more than 50 milli amps, > 0.05 Amp). If the selected scale do not allow you to have a clear reading of the leaking current and you want to change down scales, do so. Remember that some meters require to select a new hole in the meter for the red probe in order to select a lower Amp (Current) scale.
Once your meter is properly set and you have a good steady reading of the leaking current, proceed to remove, one by one, the fuses on the fuse box. Keep an eye to the meter in order to identify which fuse removal is dropping down the current in the meter. The leaky circuit is feed by that fuse.
Investigate what devices are associated to that fuse, plug the fuse back, and start disconnecting all devices one by one until you get again a reading drop. That device would be the culprit.
Javier