what is my responsibily as a bartender when it comes to serving someone who is drunk?

In most states, a bar or other commercial business commits an act of negligence if they serve someone to the point of intoxication. If that occurs, then they can be legally resposinble if someone is injured by the intoxicated person. These laws are often referred to as 'dramshop' laws. Each state has possible differences, so you should contact an attorney in the specific state.