You will realize that starting the Chapter with a character becomes more and more important as the story and series progress. Martin is a master at point of view/perspective writing so the character at the front lets you know whose perspective the events are being told from. Someone earlier mentioned that he did not know who to root for. That is the true genius of martin. Who you root for WILL change as you progress through the book. Some folks you truly despise at the beginning, you may end up truly loving and rooting for and vice versa.
He is also a horrible tease in that you will hit a point in a character's story line that is critical, and then not get back to it for a few chapters.