First you need to establish that the battery is charging. Check the terminals and the charging voltage as TB says. You should see 14 - 14.5 volts across the battery terminals with the engine running (may need to increase revs to 2,000 or so to see the full voltage).
Next, switch some load on (headlights, heated rear screen, etc) and check that the voltage doesn't drop below 13.5 or so.
Now, with the engine off and the battery fully charged (give it an hour on a charger if there's any doubt), watch the battery voltage. You should see about 12.5 volts or more with no load. Switch on the headlights and monitor it for a few minutes. It will drop maybe 0.1 or 0.2 volts but if it steadily declines to 12v or below in the space of a few minutes, and IF the battery started out fully charged, the battery is toast.
Edited to say: i agree that the radio should not drain it. A radio takes maybe 1 amp. 65 amp hour battery will power that for 65 hours in an ideal world. (OK, call it half that if you expect it to subsequently start the car!).
Kevin