2002 Chevy Cavalier Battery Good but Car is Dead

I bought a 2002 Chevy Cavalier this February/March off of Craigslist. The couple I
bought it off of seemed nice and reliable. We started the car and it ran nice. Then, a
day later, the battery was completely dead. We replaced it and the car seemed fine.
Until it randomly overheated one night. The Anti-Freeze had been leaking. I took it to a
garage and they said there was a blown head gasket. They fixed it for $700 dollars. It
was running good until recently. (Let me add that there has always been a high-pitched
squeaking sound whenever I start the car. When I hit about thirty mph it disappears. It
shakes when it gets to 70 mph). I just tried to start the car last night. The lights came on
and then I turned the key. Everything went dead. All the lights in the car went off and
nothing happened. We checked the battery and the alternator. The battery is good and
so is the alternator. We charged the battery just to make sure and then started it. It ran.
I then turned it off. Gave it a few minutes and then tried to start it again. The lights
came on but the second I turned the key, everything shut off. What is causing this? Is it
an electrical problem?