Relationship? No one can tell this person what to do but me! This way, interacting with him/her will be charged with potential conflict.
Actual play example: [game: Storming the Wizard's Tower][my character: a fisherman, father to a (secretly) pregnant shepherdess] The blacksmith's apprentice is my daughter's best friend. She knows about the pregnancy, and (out of character) says as much aloud.
Best part? I made a Charged Conversations roll, then asked, "Is my daughter actually pregnant? Is that the truth?" GM thinks about how to respond (since I was asking about something a player said, not a character. Yes, quibbling is a big part of what RPGs are about ^__^), and then says, "Yes. She is." I really liked knowing something as a player, but not as a character, and having the dramatic irony give off an awesome vibe.
Situation? I will seek the origin of the curse upon your house, and later we will ret-con that I actually placed the curse, myself! Hahaha! Now, there is another twist in the emerging situation, and it pits us against each other in a bad-blood way. Cool!
Actual play example: [game: Polaris][my character: a Knight of the Stars who is a skilled male midwife; tasked with concealing the deformed byblows of another noble house] The women of the House of Corvus, it is whispered, give birth sometimes to reptilian monsters that do not live beyond birth. It's my job to dispose of them discreetly. Then, one day, when I'm burying one in the courtyard, I discover a lizard-figurine buried in the dirt that's just stinking with dark magic. (I enter this element into the story because it seems like a fun answer to, "Why is House Corvus cursed?")
The sages inspect the thing, and determine that it is, indeed, the source of the curse. Another of the Knights (a fellow player), cynically suspecting treachery on my part, narrates, "Sir Fornax moved the curse-lizard from the courtyard of his own House into ours. He means to push his own line's barrenness onto the Corvus clan!" Ha, well, I didn't challenge this assertion using any Key Phrases, and decided it was just too awesome a twist to turn down.
Goal? I will pick the worst (best?) possible moment for my quest to conflict with yours, so as to achieve delicious drama.
Actual play example: [game: Exalted][my character: an Abyssal (a champion of death) who truly believes in the purifying power of Oblivion] My master, a Deathlord, is uninterested in anything but his own consolidation of power. When this means fighting other Abyssals instead of banding together to spread our dark power, I rebel, flee the Deathlords, and throw myself on the mercy of the Solars (champions of the sun, my people's implacable foes) rather than serve a lord without integrity.
Note - Exalted lacked any direct means via the rules (explanatory text about background and setting is not "rules" in the normal sense) for me to address theme in the way that I wished. On the other hand, if goals/relationships are not tied closely to the game system in some way (explicitly, or emergent in play), it can be tough to find your "way in" to that. My character was pretty crappy with the dice rolls in this last example, so really the only recourse I had was in my allegiances, not my actions, in play. I wasn't blocked from exploring Theme all on my own, but it would have been a lot more satisfying if character creation had gotten other players thinking in this direction, too.
Bottom line: rules can't stop you from playing as you wish, but if you follow them, they sure can help with a particular style. You can tell pretty quickly (during character creation, maybe; definitely in play) whether the game is working with you or against you. Is the game too creatively demanding for your tastes? Does it get lost in the "fluff" instead of who wins and who loses? Is it too creatively constraining? Lots of things can go wrong; playing a game that seems to "get" your style is an extra-awesome experience.