The phrase, the law of Moses, is used in a couple of ways. First, it refers to the first five books of the Bible: Genesis, Exodus, Leviticus, Numbers, and Deuteronomy. In the English Bible, the title for Genesis is "The First Book of Moses, Called Genesis." Deuteronomy is entitled, "The Fifth Book of Moses, Called Deuteronomy." Therefore, these five books are called the five books of Moses and are known collectively as the law of Moses. Jesus used the phrase in this way in Luke 24:44 - "...all things must be fulfilled, which were written in the law of Moses, and in the prophets, and in the psalms, concerning me."
However, these five books are not only law. Much of what they contain is history. The legal portion of the books of Moses is also the law of Moses. Generally, this would include the last half of Exodus, almost all of Leviticus, portions of Numbers and portions of Deuteronomy. There is not just one place where you can find the law of Moses.
Now, if all this seems confusing, you need also to consider the commandments found in this law. According to Jewish tradition, there are 613 commands or Mitzvoth in the law of Moses (not 623 as you indicate). This tradition showed up in Jewish writings between the destruction of the Temple in 70AD and the early third century. In the fourth century after Christ, the Rabbi Simlai declared that there were 248 positive commandments and 365 negative commandments.
What needs to be understood is that the Jewish scholars themselves never did agree on the exact identity of these 613 commandments. Many of the commandments are repeated in different places. At other times, what one scholar counts as two commandments may be counted as one commandment by another scholar. There is therefore much disagreement. Probably, the most famous description of the 613 commandments came from the medieval Jewish scholar and philosopher Moses Maimonides. He completed the "Sefer Ha-Mitzvoth" ("The Commandments") about 1170AD in the thirty-fifth year of his life when he lived in Egypt. I have a copy of this two-volume work in English translation and it is a fascinating window into ancient Jewish thought.
However, the Bible itself never identifies the law of Moses as having exactly 613 commandments. That is a Jewish tradition. It is almost certain that if twenty Bible students started from scratch and counted the commandments in the law of Moses without knowing that there were supposed to be 613 that they would probably come up with twenty different answers. It is an interesting piece of Jewish history, but it is not an absolute number.