I get married on May 18th 2007 in Chicago, Illinois. Our marriage ceremony was conducted in front of family and friends and the religious rites were performed by an Algerian priest in Chicago, without performing a civil marriage in USA. My question can I register my marriage with the document from the priest and the signature of all the witnesses.

The only way a marriage in the USA can be legal before the government is if you applied for a marriage license through a local courthouse and return it for certification after completing its requirements.
Illinois Marriage License Information
Your marriage license from the state is meant to be signed by husband/wife, conductor of the ceremony and witnesses. If this is the document you have, it should be submitted back to make it official.

Zenlayn,

Was the Algerian Priest acting on behalf of a church in the Chicago area. Was the priest a Catholic Priest or of another denomination?

Normally, if the priest is assigned to a local parish of the Catholic Church, his marrying you is the same as if you were married by the stat. The State recognizes marriages from within a faith community as long as the that community is a legimate faith.

I have no idea what the State of Illinois uses to certify faith communities, but a simple phone call to the local government offices will take care of that for you.