Earlier in the day, the C99 had flown from Port Macquarie to Sydney via Kempsey. The aircraft was on the first sector of the return flight. Before taxying the aircraft was issued with a 16 West Maitland Two departure clearance, at 9000 feet. The Captain was the handling pilot. The B767 was inbound to Sydney for a landing on runway 16 and was manoeuvring to remain clear of a line of weather east of the coast. The aircraft was under the control of Sydney Approach and was cleared to descend to 6000 feet. Cloud formations east of the coast between the 030 radial and 090 radial of the Sydney VOR, ranging between 10 and 15 miles, were generating clutter on the Sydney air traffic control radar. Aircraft departing from runway 16 were requesting clearances to avoid the weather east of the coast. When the C99 pilot reported ready for takeoff, an amended departure clearance was issued. The standard instrument departure clearance was cancelled and the aircraft given an initial heading of 100 degrees. After take off, when contact was established with Sydney Departures, a climb restriction of 5000 feet was issued. This clearance limit was read back correctly and set in the assigned altitude indicator by the handling pilot. The aircraft was not fitted with an altitude alert system. After crossing the coast, the Captain asked the First Officer to request an immediate left turn as cloud build ups were visible about 2 miles ahead of the aircraft. Sydney Departures cleared the C99 to turn left onto 030 degrees and soon after, for a further left turn onto 010 degrees. After advising the Captain, the First Officer selected a company radio frequency to pass details of their departure and estimated times of arrival. Whilst occupied with this task the First officer neglected to alert the Captain when the aircraft was 1000 feet from the assigned altitude, as required by company procedures. The Captain, who was not adequately monitoring altitude, did not realise this check was missed and climbed the aircraft through 5000 feet. The First Officer queried the altitude when he noticed the aircraft climbing above 6000 feet. Simultaneously, the Sydney Approach controller noticed a reading of 6400 feet on the radar display of the C99. Due to the proximity of the weather returns, the label for that aircraft was obliterated by the weather return and was seldom readable. The Approach controller immediately advised the Departures controller who in turn asked the aircraft to confirm maintaining 5000 feet. The First officer advised it was not and that the aircraft was on descent to 5000 feet. Both controllers recognised the separation standards were breached and issued instructions to both aircraft to regain separation. VH-OXE was instructed to continue climb and turn right onto 030 degrees by Sydney Departures. The B767 was instructed to turn left onto 270 degrees and to maintain 6000 feet. VH-OXE was also advised of the relative position of the B767. When the C99 climbed above 5000 feet, the minimum 1000 feet vertical separation standard with the B767 no longer existed. The minimum required lateral separation was 3 miles. The actual lateral separation was reduced to about 1.5 miles.