An Amp draw test is the way to go. Think about what electrical items work when the key is not in the ignition. Glove box light, under hood light, interior lights, light in the trunk, cigar lighter, alarm or security system, electric door locks (solenoids), clock, and maybe the sound system if it was wired hot.
If the switch or a relay sticks for any one of these it will "draw" electricity, and therefor amps.
Open the hood and remove the under hood light. Now set your multi meter for testing amps at about 20. Waite about 3 to 5 minutes after any door, hood, trunk, etc has been opened and shut if the car has an alarm. Test for amps with the meter. It is normal for the clock and alarm to draw about 5 to 15 mili-amps. If you are not getting a reading at the 20 amp setting set the meter to the next lower setting. Remember the mili-amp setting on the meter is protected by a fuse. If you use the setting and there is a greater amp draw it will blow the fuse.
If you get an amp draw reading on the meter you then pull 1 fuse at a time to see if that stops the draw. If the fuse does not stop the draw put it back in! then go to the next one. When you find the fuse that stops the draw, that is the circuit that has the problem. Find out what is on that circuit and start testing for a short, a bad switch, or a stuck relay.
Before starting any test, check all of the interior lights, trunk light, glove box light, under hood light (look from under the car),cigar lighter, and see if any of them are "on".
If you are not getting any unusual amp draw the alternator may not be charging the battery. It could be a loose belt, poor connection (loose or corroded), or a failure in the alternator. You could also have corroded battery cables, poorly connected, corroded or dirty cable connections. Check the main ground at the engine block, check the starter and solenoid connections, and the ground to the inner fender well near the battery.