I think the later versions of EAGLE has a better auto router, and whilst you can get some relatively good results with a decent auto router setup they don't take a lot of design factors into account, clock line routing for example, so I wouldn't use the auto router for anything other than a quick simple board that you can't be bothered to route yourself, BGA chips tend to house some complex blocks, not usually something that pairs well with an auto router, so I would concentrate on the fan-out and see what you can get away with constraint wise from your board house and you might have to go to 6 layers, but you'd be better off routing by hand, try a few ways of running the traces, sometimes it just "clicks".