My Rage Against the Machine.

This is a far more difficult question than you might expect. Judaism has no dogma, no formal set of beliefs that one must hold to be a Jew. In Judaism, actions are far more important than beliefs, although there is certainly a place for personal beliefs within Judaism. 

The closest that anyone has ever come to creating a widely-accepted list of Jewish beliefs is Rambam’s thirteen principles of faith. Rambam’s thirteen principles of faith, which he thought were the minimum requirements of Jewish belief, are:

1.  G-d exists
2.  G-d is one and unique
3.  G-d is incorporeal
4.  G-d is eternal
5.  Prayer is to be directed to G-d alone and to no other
6.  The words of the prophets are true
7.  Moses’ prophecies are true, and Moses was the greatest of the prophets
8.  The Written Torah (first 5 books of the Bible) and Oral Torah (teachings now contained in the Talmud and other writings) were given to Moses
9.  There will be no other Torah
10. G-d knows the thoughts and deeds of men
11. G-d will reward the good and punish the wicked
12. The Messiah will come
13. The dead will be resurrected

I personally ascribe to 11 of the 13 one this list. In that you can see that Jews all have their own beliefs about reliogion, but these thirteen beliefs are basically the foundation for Jewish beliefs.

First Establishing Who is a Jew:
The Mishnah (Kiddushin 3:12) states that, to be a Jew, one must be either the child of a Jewish mother or a convert to Judaism.

So To be Jewish your Mother Has to Be Jewish(From The Tribes of Judah, Levi, Benjamin)

Which Tribes are Jewish?:
Brief History Lesson - 13 Tribes of Israel & The Split

After King Solomon died the 13 tribes of Israel split into The North Kingdom of Israel & The South Kingdom of Judah.
The Tribes Living in The North Kingdom of Israel
-Asher
-Dan
-Gad
-Issachar
-Joesph
-Naptali
-Reuben
-Simeon
-Zebulun
-Ephraim and Manasseh

The Tribes Living in the South Kingdom of Judah
-Judah
-Levi
-Benjamin

So from this we can counclude several things.

The Southern Kingdom of Judah:
First, Judah is Jewish.
Second, Levi had already established a covenant with God and was Jewish(Exodous 32: 26-29)
Third, Benjamin sided with Judah and lived in Judah.

Judah is Jewish.
Levi is Jewish
Benjamin decided to live with Judah and Levi, who where entirely Jewish. I think that they may have converted to Judaism.

The North Kingdom of Israel:

Ten Tribes Decided to create The North Kingdom of Israel.
They were conquered by The Assyrians and Deported. They became the Ten Lost Tribes of Israel

How do you dertermine who is a Jew?

First we need to know what a Jew is. To do this we have to know that there is a Denotation (the meaning of the word) and a Connotation (what people believe the word means and eventualy it will become the Denotation) for every word.

The Denotation for Jew in the time of Judah and Israel.
Someone from the tribe of Judah.

The Connotation for Jew in the time of Judah and Israel.
Someone from the Kingdom of Judah.

Originally, the term Yehudi referred specifically to members of the tribe of Judah, as distinguished from the other tribes of Israel. However, after the death of King Solomon, the nation of Israel was split into two kingdoms: the kingdom of Judah and the kingdom of Israel (I Kings 12; II Chronicles 10). After that time, the word Yehudi could properly be used to describe anyone from the kingdom of Judah, which included the tribes of Judah, Benjamin and Levi, as well as scattered settlements from other tribes. The most obvious biblical example of this usage is in Esther 2:5, where Mordecai is referred to as both a Yehudi and a member of the tribe of Benjamin.

We can both see that the term Jew is applied to somone that is from the Kingdom of Judah. In the 6th century B.C.E., the kingdom of Israel was conquered by Assyria and the ten tribes were exiled from the land (II Kings 17), leaving only the tribes in the kingdom of Judah remaining to carry on Abraham’s heritage. These people of the kingdom of Judah were generally known to themselves and to other nations as Yehudim (Jews).
So it is seen that the term Yehudim now describes the only remaining adherents of the religion of Abraham. So it can be assumed that the Denotation and Connotation changed at that point.

The Denotation
Someone from the Kingdom of Judah

The Connotation
A follower of the religion of Judah.

After Judah is conquered the Connotation becomes more widespread because Judah no longer exsists.

The Denotation
Someone from the Kingdom of Judah

The Connotation
A follower of the religion of Judah.

In common speech, the word “Jew” is used to refer to all of the physical and spiritual descendants of Jacob/Israel, as well as to the patriarchs Abraham and Isaac and their wives, and the word “Judaism” is used to refer to their beliefs. Technically, this usage is inaccurate, just as it is technically inaccurate to use the word “Indian” to refer to the original inhabitants of the Americas. However, this technically inaccurate usage is common both within the Jewish community and outside of it.

This is my take on it, I will expand this more once I figure out how to get this thing to work.