Not sure exactly what you are referring to as the compression ring since there are a number of different headset assemblies, but you want to remove all pieces of the headset above the cup pressed into the frame. If the headset is is sealed, do not take the seals apart (I'm thinking you may be talking about a snap ring holding the seals in). Some headsets have a tapered sleeve which may be what you are talking about. This must be removed. Also, some headsets have an O-ring on one of the pieces which tends to "grab" on to the steerer tube and make it difficult to remove.

You can usually grab the fork legs and the frame and pull and get things apart.

Hope that helps and is not to confusing, sometimes it's hard to describe what I can see in my head!