A little while ago, Jay over at Life and Times of a Phillipine Gamer shot out his expectations as a player in a game. I posted then that I really admired it- it is far too often that players do not know what they want from a game.
In certain kinds of games, this is ok. I'll say that 4E D&D and Pathfinder offer the option of waiting to see where the adventure takes you. Those games are (as a rule) pretty prescribed into "what do you DO" and not "how do you FEEL".
Other games, it really pays to have some concepts of what you want to do with your character (other than show up to beat on things). Having at least some character development goals can immensely improve your gaming experience. It can enrich the group setting and help everyone have an idea of 'what to do' when moving from place to place in the game.
It is important to talk to the GM and be sure the kinds of expectations you have match or mesh with the kind of game they are running. If you are in a gritty, dark & dirty near future game of some sort, wanting to become a porn star is probably feasible. If you are playing a silly dungeony old school variant, wanting to become a serious magical theoretician is probably not going to work out so well.
In addition to the kind of game, it helps to know that the GM is into the same level of "work" that you want. In one of the games I watch, the GM does a TON of work. If I walked up to him and said "hey, I want to do a bunch of private scenes to help with character development", he would at least consider it. He might say NO because that's adding to his workload, but at least it is in his realm of possibility. If, however, I did the same in Jeff's amazing dungeon game, he would almost definitively say "no". That sort of thing doesn't mesh with his style of running a game, overall.
Overall, I would hope that I'd know before I ever joined a game whether my playstyle and the GM will mesh. I would HOPE that I would investigate and ask questions, and make sure the GM did the same. But it's happened to me more than once when I have gotten in a game where things just didn't work, and I didn't know how to get what I wanted. I've put that knowledge to use, and have learned for the future, but when you're in the thick of it, it can really suck.
So what do you do when you want something the GM can't or doesn't want to give? How do you handle wanting intense, deeply personal private scenes with a GM that overschedules himself and barely sleeps, much less has time for "bluebooking"? How do you handle wanting to show up to "beat on stuff" with a GM that really wants character development hooks and rich, thick and immersive play?