Simply, the object of a bad weather circuit is to enable you to remain visual with the landing runway whilst completing a circuit to land. The assumption is that either the cloudbase is low or the the inflight viz is poor or both.
Therefore the circuit will be flown much lower than usual and you will maintain a position far closer to the active than usual. You must not enter IMC and in "real" conditions will be adjusting what you do to avoid doing so. In reality how low and how close you keep the circuit will depend on the conditions on the day and how comfortable you are with the aircraft. When and where you take flaps, undecarriage etc will also vary depending on you and the aircraft but the circuit will happen a very great deal quicker, the turns will involve considerably more bank and final will be very short.
I doubt you will be asked to demonstrate a bad weather circuit but you never know. If you are, 600 feet would be a sensible circuit height, unless told otherwise, setting up for crosswind over the numbers (unless the runway is very long), keeping the runway within your wing downwind and turning base to keep final as short as possible. Dont forget to ask ATC if you can fly a low level circuit (it is for them to grant) or if AG make sure everyone else knows what you are doing.