Q: How can I see that there is no "ME"?
A: The question itself assumes that there is a "ME" to see that there is no "ME". So as an intellectual exercise it is always going to fail. We generally approach this question by looking and trying to make sense of present experience in a different way, of waiting for experience to change somehow. Yet we don't really want to dig deep into this idea of "ME" - find out what that means, find out what it's based on.
"ME" has a very deep and strong foundation. It is the idea that what you are began at a certain date and time. It is that entity which is "ME". And that "entity" isn't a tree, it isn't the desk, it isn't the wall. It IS the body, it IS the mind, this is what we believe to be the case.
So it is this most basic and deeply-held belief about what we are which asserts that you are separate absolutely. THAT is who is trying to see that there is no "ME". That entity. But because that entity is false, it cannot find anything. It is an idea trying to eliminate another idea, or an idea trying to eliminate itself.
The spiritual search is very conceptual. Just look into the idea of separate existence itself, look into the idea of things, the idea of matter, the idea of energy. See what is the essence of each "thing" that we arrive at via some means of knowledge. We find that we are always chasing something which never appears, something which is the essence of what appears to be yet we can never pin it down. We find that existence is never arrived at, we are only ever left holding onto forms OF that essence, appearance OF that essence, like trying to capture running water in our hands.
The idea of separated existences is found to be false, therefore the "ME" was always false. It didn't need to be eliminated because it was never there, it was a false idea, it was a false assumption. So we didn't get RID of "ME" - the bottom fell out of the idea.
And therefore what IS was always IT, already. What we know AS "ME" is then all there IS. Or as Shankara put it, there is ultimately no difference between Brahman and Individual Self.