1. Remove the starter, solenoid and battery.
2. Charge the battery, leave it overnight.
3. Take the alternator to a shop and have it tested on a growler.
4. Apply 12 volts to the soleonid. Do the contacts close? if so it's likely good.
5. Measure the resistance across the relay when the contacts close. It should read zero ohms. If not, replace the solenoid.
6. If everything tests OK, re-install it.
7. There'as RED wrie ( I think) running from the alternator up to the common positive voltage distribution lug on the solenoid. Does this wire read zero Ohms of resistance? If not, it should, replace it.
Here's a rule of thumb I go by when jump starting a vehicle:
Fast charge to start = Problem is Battery.
Slow charge to start = Problem is Alternator.
The tests on the alternator are free.
Start there and come back with the results...