A note: While this is a system-agnostic question, certain systems (ex: DitV, FATE, Paranoia) are much better at handling this than others (ex: Any D&D system). Some games are even focused entirely around CvC conflict (En Garde, Everyone Is John, etc). For the purposes of this response I'm going to assume that in this game the party is all on the same 'team'.
As usual, my first recommendation is that you make sure everyone wants to play the sort of game you're after. Sit down with your players (as a group or one-on-one) and see how they feel about the idea of inter-party conflict. The beginning of a campaign is a FANTASTIC time to discuss changes in tone like this, especially if you start one in a system that encourages inter-party conflict. Encourage openness here. If they all seem to want to try it, continue. Otherwise, I recommend you back down, because trying to make this sort of game happen with the wrong group of players is an excellent way to hurt peoples' feelings.
Now that we've got everyone on board, the first thing to do is explain to your players that it's OK (and in fact, encouraged) for characters to have arguments, disagreements, romances, etc. Many players, especially in mono-gendered groups, will likely feel uncomfortable with the idea of pursuing another character in a romantic fashion, so you may wish to leave that off the table until they get a little more comfortable with these forms of role-playing. Again, if players seem reluctant, back off.
But if your players are up for some conflict, a good way to start off is by encouraging PCs to keep secrets and make autonomous personal choices.
There are two types of player with regards to secrets. The first type prefers to know only what their characters know, to be able to plot on their own and pass notes with the GM. The other kind prefers to be completely open out-of-character and only keep secrets in character. Naturally there is a third type wherein no secrets are kept, but we're assuming against that for this exercise. Try to get everyone to use the same system. In my experience, it is easier to get everyone to have a good time and cooperate to make shenanigans fun when everything is out in the open for the players, but the characters keep secrets from one another (ex, you might tell the ranger, "You see the pixie messing with the paladin's blanket!" in front of the entire group, and leave it up to her to decide how to proceed). This helps to make everyone feel like they're in on the joke, and allows characters who are being tricked to have fun 'playing dumb' and going along with the gag. This is definitely the safer bet if you're afraid of people's feelings getting hurt.
Encourage the players to leave 'hooks' in their characters, especially ones that lead to something they may want to avoid telling the other party members. "I'm looking for my long-lost father" is an OK character hook, but "I'm looking for my long lost father who I last heard was involved with the Evil Empire" is more along the lines of what we're after. Regardless of how interesting you find your players' hooks, be sure to bring them into the campaign in some way. Work with your players to make their stories and hooks work with your setting. Feel free to entwine the characters' back stories, especially in ways that create conflicting interests and goals. These back stories, and the way they bring the character's personal journey into the focus, will help not only to create conflict, but to make the PCs and the world feel more alive and fleshed out.
What you will (hopefully) start to see is a divergence of party objectives (assuming your party is not entirely composed of Lawful Good Paladins of the Righteous Truth, in which case you may wish to re-think your campaign goals). Previously, the entire party's objective was to 'win', but over time, keeping secrets and personal goals should naturally branch characters into a number of side objectives (don't let the party find out about my stealing from our employer, redeem my father even though he killed my friend's son, hide my alcoholism, etc). Now that we've decoupled the party objectives a little, we can sprinkle in some moral choices.
Autonomous Personal Decisions
As you're designing encounters, feel free to leave situations which are morally ambiguous, or in which the party is likely to be divided. If the paladin goes along with the party's decision to loot a tomb, or if the by-the-books cop has no problem with the party's decision to lure the big bads into a large open area filled with civilians for the big shoot-out, feel free to ask them if that is what their character would do. Likewise, feel free to encourage the pixie to play practical jokes, the scoundrel to be a scoundrel, and offer the hitman the occasional 'job' to help pad his bank-roll. Allow your players some autonomy from the group's decision. While splitting the party can get very messy, it is OK for characters to do their own thing every now and again, or for some characters to sit out from a plan they aren't comfortable with.
If someone does something remarkable, awesome, or highly in-character, award role-playing XP. Did the high-strung businessman just express his concerns with the brutish thug's methods? That's worth some role-playing XP. If the thug reacts in character, feel free to toss him some, too.
So now we're all acting of our own accord. How do we keep things in control?
Again, everyone needs to understand the difference between the PCs and the players. Make sure the group is clear on that.
When somebody starts shenanigans, look around the table. Is everyone laughing, or is someone uneasy? If players are uncomfortable, especially if a player feels like she's being picked on, it's time to either talk it out as a group or drop this play style.
Finally, make sure everyone has their time to shine. Backstory missions are awesome at making people feel important, especially in small groups. Make sure everyone gets a turn at doing something their character cares or feels strongly about. Make doubly sure that each player feels like he or she is contributing both to the story and the party. And finally, ask for suggestions, comments, etc. after the session. This is a great time for the players to bask in how awesome they were, or to point out parts of the game they loved (or hated!), what you should do more of, and what you might want to lay off on. Try not to take things too personally; if a GM reacts well to criticism, players will feel free to be honest, and everyone will have more fun in the long run.
Most importantly, have fun with it! Laugh with the players when the entire party is embroiled in a terrible snafu, or when the halfling gets caught shoplifting and spends a quarter of the session running from the cops! We're all here to have fun, so go with it!