You sort of spoke my unspoken thoughts, lara1. Maybe Liz's "purpose" is to eventually assassinate Red? Whether she is his daughter or not doesn't matter, except in his own mind. And what if he has known all along that he has to kill her in the end, if she discovers the truth about her purpose (discovering who she really is might lead her to that discovery as well)? But this probably goes down to what is Red's true purpose and role in whatever it is that he really does. In fact Liz could be the one to,take down the person in Red's place not necessarily Red himself. He just happens to be the one that is in that position at the moment. Ironic that his own child would end up being the one set up to remove him. In the game, whatever the game really is. I am thinking that the people are not important, it's what they represent, their positions, that are what counts. Probably it was never intended that close relatives might end up in those positions. So if that's the case, then the master strategist is probably working to,prevent the game from playing out and being removed by his own daughter. Checkmate.
So then we have one of the other players, Kate,who by trying to,protect him, ends up making it worse. Or else maybe her purpose is to make sure the game plays out as it is supposed to. That is assuming that it is inevitable, which it may not be, it could be very fluid and flexible for all we know.

Or Red could simply be a criminal mastermind and none of this other stuff is really even part of it. 

All great ideas! I still personally want Kate & Dembe's numbers to follow one another because it fits better in my bubble . For all we know their could be yet another organization or big bad who is the true controller of the blacklist and assigning numbers/ranking/positions (whatever you may call it) and Red has been hired to deliver them.