Who was genesis written to?

Genesis was originally written to the Jewish people by their great leader Moses, to tell them their history and the history of mankind.

But Genesis is also written for our benefit to tell us of our origin and about the Creator and His promises that He keep with mankind.

Another Answer:

Genesis, as is all Scripture is for all mankind. The descendants of Abraham, called a Hebrew, were called the House of Israel. The term Jew(ish) does not come into play until the Babylonian Conquest of the Southern Kingdom of Judah:

2 Timothy 3:16New International Version (NIV)

16 All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness,