Doc here,

imp:
Check your ground buss system FIRST..
GROUNDING:
Run a properly gauged Ground cable from the battery to any handy bolt at or near the starter, From that same bolt get a Battery cable at the parts store, that has two 3/8 ring terminals on each end and attach it there and to the Frame.
Next get some 10 gauge wire and ring terminals and run two wires from your bolt on the block to the Firewall, and the other to the Alternator bracket or mount bolt.
Next get some Wire Braid, (Radio Shack) and install braid from the radiator support to the frame, Fenderwells to frame , hood to firewall, Doors to door posts, gas flap to body, tailgate / Trunk to body.
At each point the wire is grounded, Burnish ALL the paint and grease off to bare metal. Use a proper star-washer and lock. Use sheetmetal or Tech screws where no screws are available.
It sounds like a lot of work, but after you assemble all the parts, it's only a few hours to do..and you'll end up with a system that will work reliably for many years to come..and can eliminate that from your troubleshooting list.
If that does not fix the problem, Be SURE the belt is tight, and the connections at the alternator are secure and CLEAN.
Next, disconnect the battery cable, and check the cable at the starter solenoid..Be sure it is tight, clean and the solenoid or wire is not melted or burned..
A CLUE HERE would be you installed a 1 wire alternator, and I assume it went to the battery terminal on the solenoid..(loose or poor bonding)
Have the solenoid tested..a freebie at auto zombie. A quick test would be, after duplicating the problem, jump the "S" terminal to the battery terminal, if it cranks, your wiring is at fault, if not, the solenoid (or even the starter ) is at fault..If it "clicks" out real quick but turns slow, I'd pull and test or replace , OR rebuild (cheap and quick to do with a kit..) the starter..may have bad brushes or burned armature contacts..
NEW battery's fail too. Have it load tested.
Doc

imp: