'Game Player' implies that a persons actions and words are disingenuous, not to be trusted, and are said and done to advance a particular objective, to whit making it to the final and winning the prize, and are therefore 'false'.
Insofar as Big Brother is, ultimately, a Game Show, it shouldn't be a surprise that people should try to 'play' the game to improve their chances of winning, however by extension nobody likes to think that they have been 'played' and are the victim of some manipulative strategy, hence the opprobrium assigned to the term 'game player'.
Big Brother is, if you really think about it, a very odd 'Game Show' because the level of audience investment and audience interaction (even now, when producers have ****ed about with it beyond all reasonableness) is such that someone who is perceived to be consciously manipulating other HMs (and worse, the audience) is given very short shrift indeed, but there are still ways it can be done.
The difficulty is that any house mate needs not only to 'play' their peers, but also the audience, but do so in a lighthearted and smart way, and more often than not they need to play or at least play along with 'Big Brother' themselves, all without coming across as too Macchiavellian, and it is very difficult to get the balance right. You might sail under the radar with your fellow HMs as they expend all their energy butting heads with each other, you might even bring the public along with your plans, as to a certain extent Ashleigh did during Girl Power week, but it is very difficult to keep up both angles of 'attack' and there is still the risk that BB will decide to target you and expose your worst moments to the public and the other HMs and screw with you.
There is plenty of evidence that suggest the best strategy for winning is to keep your head down early on, roll with the punches, approach tasks with enthusiasm and humour, be selective about when and how you "fight your corner", be wittily but constructively critical of others (esp. in the Diary Room) and be more assertive in the final 3 weeks.
Seems simple enough, but it's a lot to remember and hard to actually put into effect with practical actions, particularly when BB and the experience in general is messing with your head!