If I am not the skandhas, then what entity/group of entities am I being responsible for by being a good person?
Buddhist concept of not self is always difficult to translate. It generally means:


There is no core or soul
You have no absolute control over your aggregates or the world at large


Also when you have the perception of self you have a yard stick which you use to measure other people, their actions, standing, wealth, etc. with respect to you. This creates misery. If you drop the yard stick then you get out of misery.

Having said this though there is no lasting entity as you there is a mind matter process. What ever you do becomes fabrications which at that point influence the process and also has a "loop back" at different times.