Sorry for repeating myself, but as you can see on the nLab here, Colombeau himself has written an elementary introduction to his theory, mainly for people who are interested in applications:
Jean François Colombeau: Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics. See this review in the Zentralblatt Mathematik. This will also give you a hint if this approach is suitable to your problem (BTW: what is the application you are thinking about?).