Standard procedure for spreading satellites within an orbital plane is to change their altitudes, and hence their periods (how long it takes for them to travel around an orbit). Once the altitude changes, the different periods will start spreading out the satellites around the orbital plane. Once their desired position is achieved the satellites are all brought to the same altitude. Their period is now the same, and the separation is roughly stable. Ongoing corrections to the orbit to account for perturbations will be needed, however, to keep the exact spacing.
Traditionally the change in altitude (and the ongoing orbit maintenance throughout the mission) is done with thrusters. There is a trade to be done, however. Changing the relative altitudes by a large amount will cause the relative drift to be faster, and therefore final positions will be achieved sooner. However the larger changes in altitude require significantly more propellant.
An interesting alternative is to vary the ballistic coefficient of the satellite to control changes in altitude. This can be done by feathering the solar panels, for instance. As more surface area is exposed to the "wind". drag increases, causing the satellite to drop in altitude. Each satellite within the plan can be made to drop different amounts by controlling the ballistic coefficient, and this in turn will cause them to spread out. This was first demonstrated by the ORBCOMM constellation. See this patent.
Note that the same problem in reverse is solved by vehicles attempting to rendezvous with each other. For instance the visiting vehicle to the ISS may start in a lower orbit, hence going faster - catching up and potentially lapping the ISS. Its orbit is slowly raised to match periods with the ISS. The timing is done so that when the period is perfectly matched, the two vehicles are very near each other.
Changing altitude/period only directly addresses spreading the satellites within the plane. Getting multiple planes in the constellation is normally done through different launches (one for each plane). However, it IS possible to spread out the planes in a similar fashion, if you are willing to wait long enough. The precession of the right ascension of the ascending node (RAAN) [or "swivel" of the orbit" is dependent on altitude. Different altitudes will result in different precession rates, slowly spreading out the planes around the earth. However this process is measured in months/years and usually not cost effective.