Going to state the bleeding obvious here, but you need a battery charged from a different charger to prove where the fault lies
Never had a charger go wrong, but first time for everything and very unlikely both of your batteries have died at the same time
So going for camera fault, have you updated the firmware lately? just a thought. The DC-DC board went on mine, its basically the power controller plus fuse, not an unknown fault and would be my best guess. Canon at Elstree do a fixed fee repair for 153 quid which includes a service and sensor clean, work guaranteed for six months too
Could try taking the battery, memory card and date/time cell out, leave for a while then give it a go. Sometimes the date/time cell goes and causes some unusual problems, might be worth changing that for a new one