How did the first self replicating organism come into existence?
Evolution or (as Darwin called it) "descent with modification" is a theory which explains the origin of the species NOT the origin of life.  How the first life arose is completely irrelevant to the theory of evolution.  What evolution does explain is how and why we have such variety of life on earth all descending from the same organism.

What you're asking about is not a theory of "evolution" but rather a theory of "abiogenesis."  Although there are many interesting hypothesis for how abiogenesis happened (e.g. the RNA world, the "metabolism first" theory, etc.), the fact is we simply do not know yet how life first arose.  What we do know is that life first arose between 3.5 and 3.9 billion years ago.  That's a really long time ago compared to lots of other important events in natural history (even the Cambrian explosion where modern animal phyla evolved was only half a billion years ago), and so it shouldn't be surprising that it's a hard problem.