What is the purpose of the "Giant's" Game? How should Ender evaluate his success at this game? Is he a murderer?

The "Giant's Game" was a test, a way for Graff and the others to guage Ender's capacity to problem solve, deal with stress, and most of all his eagerness to win. Graff needed to know that Ender could perform under pressure, and that he would be willing to do whatever it took to win.... even if he had to break the rules.