No, normally a person is not found to be at-fault in a car accident because they did not have a valid license and were driving without insurance. If a person is unlicensed and uninsured he or she should be cited by the law enforcement officer that responded to the accident scene for these violations of state laws. Fault will be determined by your insurance company if claims are placed with them.
The fault for the accident though will depend upon what happened with the vehicles, placing the person's violations of licensing and financial responsibility aside. For example, if you run a red light and hit a car it will still be your fault for failing to stop at the red light no matter if the other driver, going through the green light, is uninsured. It might be frustrating that the other person was illegally driving but they should be ticketed for those separate offenses and face the penalties related to those violations.
In most cases you are also responsible for the damages you caused the other party if you are found at fault. There are some state, such as Louisiana, that have "no play, no pay" laws. In Louisiana this means that drivers who do not have the Louisiana mandatory $10,000 liability insurance forfeit their right to use the court system to sue for the first $10,000 of certain types of damages.
For information on your states insurance laws and if you will be responsible for the other person's damages speak with your insurance agent and/or contact your state's insurance regulator.