Self inquiry is the implicit or explicit process of discovery, by which one uncovers a deeper understanding of who one is.
Implicit self inquiry happens continually in everyday life, even if you have never engaged in any deliberate attempt at self-realisation.
Explicit self inquiry occurs when you deliberately contemplate or meditate with the purpose of gaining insight. It is just such self inquiry that discussed here.
The premise of self inquiry is that one may possess some misunderstanding of their own fundamental nature. As such, it is important to remain open to the possibility, to some surprise or misapprehension.
In this vein, thought alone is inadequate to the task. Thought warrants special mention, because it is often tightly associated with our sense of self. Thought is a useful tool to structure and organise your approach. It can even reflect upon itself to a certain degree. Fundamentally, however, the essential nature of self is beyond thought. Be open to the possibility throughout your investigations, otherwise you are defining yourself before you even begin.
Self inquiry is not about reaching alternative states of consciousness, though such states may occur at any time. It is about seeing yourself, and reality, as it really is.
Historically, a diverse variety of methods have been advocated for the process of self inquiry. For instance, Nonduality teachings often begin at the level of thought, and continue through the felt exploration of bodily sensations. Meditation, contemplation, yoga, prayer, service, creative expression, journalling and philosophy have all been applied in this pursuit.
In the simplest sense, I would suggest only two considerations. First, that you keep the essential question nearby as you engage in daily activity - Who am I? Second, that you regularly take time to pause, relax and observe as-if an impartial bystander to the entire subjective experience, particularly your thoughts. Consider it an informal meditation, to be conducted upon every waking moment of your life. Though if you return to it regularly, you need not exert yourself in any way. Your subconscious mind will continue to watch, even when you do not appear to be present.
Whilst it is possible to answer such questions quite quickly, such answers are unlikely to be fruitful. The first answer is unlikely to be the last. Penetrating insight does not occur on your timetable. Be open to the possibility that there may not even be an answer, in any explicit sense!
For a more energetic practice, engage in any number of activities that bring your awareness-intelligence to the exploration of your nature. It could be meditation on the meditator, focused intellectual inquiry into the I, or creative exploration of a felt sensation or feeling.
In all such practice, you will possibly gain most benefit by maintaining a focus upon the root concern. You are cultivating an open curiosity toward the experience of being. As if closely examining the lines on the palm of your hand, the patterns in the clouds above your head, or the sound of your thoughts. When you pay attention to anything, your mind is quieter. Seeing happens by itself. Always the same question - What am I? - whether it be explicit through mental cogitation, or implicit through your attempts to convey a feltness upon a canvas or within a piece of clay.
There are many expressive modalities which may be very useful in this pursuit. Poetry, drawing, painting, singing, dance, humming, drumming and sculpture to name a few. Exploring difficulty in this way can be particularly cathartic, and may lead to significant insight.
Engaging in creative expression helps shift the mind from narrow, analytical, piecemeal thought to wide, intuitive, holistic conception. If your mind is quiet, you are probably using the holistic perspective. These are two superficial facets of your nature, and two tools for investigation. Thus it is important to use both.
A process I have often found useful is journalling. Careful, patient and meticulous journalling of your present understanding of self, of what is true, and what is known, is a wonderful way to engage the pregnant silences as well as the usual verbiage of mind. Many of my insights have occurred in this way.
There is a tendency to make this process more difficult than it needs to be. With so many traditions and so much written material upon the subject, self inquiry and self-realisation have become associated with the esoteric. Whilst an appropriate characterisation in a certain sense, in another way it is quite misleading.
Without realising it, everyone brings with them certain assumptions, expectations and beliefs about the nature of self, and the outcome of any self reflective process. Unfortunately this baggage tends to trip us up at every step.
Self-realisation is not far away. It is simply you, here and now, coming to a greater experiential understanding of the essence of whatever it is that you are.