Young man: This period called youth has been always a very dangerous age throughout human history all throughout the world. Even the world-famous poet and dramatist William Shakespeare accepted the unusual behaviour of youth when he describes youth in is poem "Crabbed Age and Youth" as youth being "a summer morn, a summer brave, full of sport, being nimble, hot and bold, and wild."
Your age carried you through to the stage when you went on a wild adventure of driving a car without a license, that too as you state "drifting to enjoy", thus endangering the lives of others. Furthermore, the hot blood and the boldness of youth in you made you to take a stand when you were caught doing something illegal, and was finally made to face punishment. Now that you have "hopefully" grown up a bit, you are possibly getting to feel the repercussions of your wild act.
Every country has its own rules and regulations. Rules differ for the country's own nationals and for foreigners. Since you are foreigner in Qatar, it may be the law here that once a person has been involved in such misdemeanors or misadventures, they must be sent back to their country. As such you have been served with the deportation notice.
It is very difficult for any one to advise you on how to get your name removed from the blacklist. I doubt very much even lawyers will be able to help you out on this but then you can give it a try by consulting someone in the legal fraternity. The only point through which they may be able to make a request to the government is your age when you went around on a driving spree.