If the overnight charge doesn't help, then the battery is garbage. That is of course assuming all of your connections are fine. If your battery is charged you don't need an alternator to run (although you might have some strange electrical behaviour, dash lights and such, due to a bad alt - but the motor should run fine).

An important point to note is that a dead battery subjected to freezing temperatures will freeze and crack internally, and it will no longer hold a charge.
Also, a battery should never be stored on a concrete surface.