First of all, a circling approach IS a visual manouevre - whilst the minima for Cat C aircraft is fairly generous (2400m vis), I wouldn't fancy doing one in those conditions. If visual contact is lost at any point during circling a go-around must be flown.
As to actually flying the circling part - in a modern jet airliner like the B737 or A320 it is not sufficient to just 'eyeball' it- we need some numbers to help us manage the momentum and position of the aircraft in order to acheive a stabilised final approach to the landing runway.
So, in the 737 (my current type), to land on the runway reciprocal to the instrument approach:
Approach is flown to circling minima to the instrument runway - if an ILS approach, NOT using 'APP' as it is impossible to disengage after G/S capture without taking the A/P & FDs out. So HDG SEL and V/S, and level off at circling minima by pressing ALT HOLD at (approx) 10% of the vertical speed above the circling altitude.
This part of the approach is flown with Gear Down and Flaps 15 ~ approx 150kias. Autopilots in until the final turn.
Then a level segment is flown to 1/2nm from the threshold of the instrument runway, at the same time the MCP is reset for the missed approach altitude.
At 1/2nm from the threshold, we turn left or right (depending on airport restrictions eg terrain, ATC instructions and/or pilot preference) to give us an offset TRACK of 45 degrees (i.e. heading corrected for wind). Start the stopwatch.
After 45 sec (wind corrected, 1 sec per knot of head or tailwind) turn to track parallel to the runway - downwind to land.
Abeam the threshold re-start the clock.
After 10-15 sec (wind corrected):
Autopilots OFF
Flight Directors OFF
Flap 25, 140 KIAS
..and start a descending turn onto finals.
As we turn onto finals, select Flap 30/40 and the appropriate approach speed.
Aircraft should be stabilised (configuration, speed, rate of descent) by at least 300' above the landing threshold.
Land.
Of course, if the landing runway isn't 180 degrees out from the approach runway, things are a little different - but the objective is still to arrive downwind to the landing ruway at about 1 1/4nm spacing, at a typical circling minima of about 500-600' QFE.
Hope this helps, happy to expand on any terms you are unfamiliar with.
Profile for the A320 is very similar, but the automatics are slightly different.
Whoops! Ibanez beat me to it. He has some good gen in there which I'd forgotten, about adjusting the timings for variations in circling altitude. I don't fly circling approaches very often (thank God) so am a bit out of practice!
The above taken directly from our Company Ops Manual.