What is the problem when 1998 Chevy makes clicking sound but won't turn over then when battery is charged car still cuts off is it the alternator?

It sounds like the starter is bad. The clicking you hear is the solenoid trying to kick-over the starter.

If the vehicle starts but, won't stay running, it could be the alternator, or the wireing to (from) the battery or alternator (most likely the ground wire). Or it could be the dreaded MASS AIRFLOW SENSOR!!!!