Jumpstarting...can it kill alternator?

Once I took the batter out ( a pain in the butt because its in the fender) I took it to Advance for them to check it and that is what they in formed me about the amps but said that the reading is saying that its chargeable so I left it there over night. Then I picked it up the next day and they read me a receipt that it was fully charge. I then clean and check the cables and made sure the posts didn’t have any rust and at first there was no Check Batter light but after driving a few minutes (Less than a mile) the light came back on? I then drove back home, turned the car off and tried to start it right back up and it wouldn’t start or even turn over? After a half an hour I tried to start it again and behold it started and again no Check Battery light but I’m sure it would have came back on so I just turned the car off and I’m looking for some type of help and I can’t afford a alternator and a battery so I’m trying to pin point what is the correct problem.