If it charges and runs fine during the day then your charging and battery systems are fine. You have a large power leak. Something is being left on all night. Use that same volt meter to check some things that should be off when the car is off, you may find that the ignition is mis-wired or is not shutting off all the way. also, have you installed any big audio amps or anything electronic lately? Check for power on those when and where there should be none. Worst case, just pull all of your fuses one night and see if it starts in the morning after you put them all back in. If that works, then try them one at a time, over however many nights it takes to go through all of the fuses. Then again, you could also check for power at the fuses when the car is off. Unfortunately some should have power at all times, but most should not. Just use good sense... the clock will always have power, but the wipers just don't need power at 4am; stuff like that. I'd check at the fuse panel first.
Good luck
Johnny