As championed by Ramana Maharshi, self-inquiry is the simple practice of asking, "What am I?" as often as possible. That's it. Nothing complicated. Take this question as far as it will go, and then keep going. Really discover, for yourself, who you really are. Find the core of you that never changes, the core that has been there throughout your life. Don't stop asking this question until you get to that core absolutely.

Other than that, there really is no technique to self-inquiry and not even a set method of how to ask that question. Ramana also gave the practice of "Who am I?", and that is valid, but has a slightly different target. Both are fine as long as the goal of really getting the root of who you are is pursued ruthlessly.