First of all, charge the battery up for 8 to 10 hours at about 10 amps..Then take it and have it load tested at the local auto parts store..If it's bad replace it..
Next if the battery is good and CHARGED fully, Does it now start/run? If so, turn everything on, power wise and measure the voltage across the battery running..if less than 13.75 or Ideally 14.4..you have a charging/Regulation problem..
Being as you are getting a low / dim Battery lamp and a buzzing relay (usually the choke relay, but could just be lights on buzzer) Sounds like you have some dead diodes in the alternator..you can Replace the diodes yourself or just replace the whole alternator, ... your choice based on your skill level and tools..
Have the alternator tested when you do the load test..
BTW, these are usually free at the local auto parts superstore..If the alternator tests good (which I doubt..) then you have grounding problems...
Your ground should be set up as such, The battery cable ground to any handy bolt at or near the starter, from there, get a battery cable that has 2 3/8 ring terminals on it ($6 bucks at wally mart or auto parts place) and ground that to the frame..From the engine block ground at the starter, run 2 #10 wires, the first to the firewall, and the second to the alternator ground lug or mount bolt...
USE proper Star washers, Sheet-metal screws / bolts and locks, and burnish ALL the paint and grease from the grounding areas..this will give you a rugged reliable grounding system that will most likely outlast the car...
If the Alternator is powered white Along with the mount, remove both and the hardware, and sand or water blast them back to clean bare metal...then when you reinstall add aluminum wire bonding paste to all hardware and anywhere the mount or alternator touches the block ..
This is electrolysis , the mount is aluminum and he block is steel (what the hell were they thinking?)..that powder is an insulator..
It will ALSO dissolve the water pump impeller like a mild acid..so you might want to think about replacing that while you are in there too..(less stuff to pull down again..)
If all of that dosn't solve the problem..you may have ONE out of 3 fuse links burned off the main lead..(less likely though)