1970 Mustang Electrical Problems

Try disconnecting your + lead from the battery and securing one end of your volt meter to the battery post and the other to the + cable lead. This will give you an inline indicator of voltage draw. From this point start unpluging stuff one at a time and you should be able to locate the short. I would suspect your alternator harness if the engine dies while it's running. I hope this helps and good luck! If it does invest the $ in stang parts.

Is the 70' alt internally regulated? I can't remember if it has an external voltage regulater. That could be a problem. If no external reg, then what is the voltage at the battery with the car running? Should be 13 and a half to 14 and a half. Should get over 14 when you rev it up.
If all the parts are good and still no charge, check the field winding. One easy way to check this is to "carefully" place the tip of a metal screwdriver on the back of the alt in the center. THere should be a magnetic attration. If there is, that tells you that the field winding is energized. If not, check the field connection on the alt. If you are not getting power to the field winding, it won't charge. Found a diagram
http://www.472ci-67mustang.com/wiring_diagrams.htm
SHows an ext reg. And the field (F) wire is Orange. GOes from the F terminal on the regulater to the F terminal on the Alt. Check this as well.
If all that is good, hook your meter in series like GoinBroke said and find whats killing the battery. You can also put the meter to measure current hooked up this way. THere should be near zero current draw with everything off and the doors closed. Make sure you defeat the door buttons so the interior lights don't mislead you while pulling fuses to find the circuit that is drawing the current. Hope that helps.