You have to determine and define "completely dead." Could be as simple as a bad battery clamp or an internally corroded battery cable.
Get a test light, multimeter, and a bag or two of clip leads from Radio Shack. You should have these anyway.
Stab the light/ meter right into the top of the battery posts. Anything there? Turn on the headlights -- any light, glow, anything? Does the test lamp go out or the voltage reading drop?
If the battery shows some voltage, even 8V or so with lights on, but the lights don't work, move your meter/ test lamp to the two battery clamps -- Any test lamp, any meter reading?
Now move your test lamp / meter to the engine block, and to the "big stud" on the starter relay. Any reading with headlights on?
If not, the trouble is right in that area --dead/ bad battery, bad clamps or cables, etc
If you do have at least 8V or more, and no headlights, could be fuseable link or bad connections through the bulkhead connector, or bad connections at or even a bad ammeter.
What do you have in the way of a shop manual or wiring diagram?
Wiring diagrams from MyMopar somewhat useful, not always complete:
http://www.mymopar.com/index.php?pid=31
Direct links. Far better than nothing
http://www.mymopar.com/downloads/1964/64ValiantA.jpg
http://www.mymopar.com/downloads/1964/64ValiantB.jpg
I don't know how much help, here's a free 65 Plymouth service manual. Electrical type in page 8-2 in your viewer, then back up one page to 8-1. Wiring is on page 8-96
[ame]https://www.dropbox.com/s/upks33txhbe9tjw/1965%20Plymouth%20Service%20Manual.pdf[/ame]
This MAD electrical article has a simplified diagram that shows the main power path. I believe that in your car, the main feeds do not go through the main bulkhead connector, but rather have their own separate feed-throughs off to the side
http://www.madelectrical.com/electricaltech/amp-gauges.shtml