We are used to perceiving the physical world directly and infer that Consciousness as an aspect of the physical world. In meditation this can be reversed: one can experience Consciousness directly and perceive the physical world as an aspect of Consciousness. It is also possible that the two are independent of each other or that they are one and the same.
Because Consciousness is, by definition, the Subject, it can't be directly observed. It's is like an eye trying to observe itself. It can only be directly experienced. In fact, in a very real sense, it is experience. For this reason, observing Consciousness is problematic.
However, the experience of Consciousness can be intensified by removing the activity of observation as much as possible. When no objects are being observed, the only thing left is Consciousness. In this state, Consciousness still can't be observed directly but the experience ("awareness") of it can be very intense: Consciousness becomes conscious of itself.
Try the following. It’s o.k. to have thoughts; let them come and go. The goal is to have few, if any, thoughts, but still retain awareness. It is very possible to have much reduced activity, until there is only the occasional thought ("Now I should try to be aware of my awareness.") and periods of quiet between them.
Our identity gets built from the things we can observe about ourselves. Because our Consciousness can't observe itself, our identity gets built from things which we specifically aren't -- i.e. all the things that are external to our Consciousness.