Inciting intrique (without forcing firefights)
- Ensure that both factions' real goals require resources or skills that exist in the other half of the party. If the goals can be accomplished alone, the party splits into two separate parties that merely happen to be travelling together.
Example: If faction A contains the party mage, make sure faction B has some goals requiring identify spells, or detection of magic, or where a fireball would be handy. The intrigue comes from trying to create plausible reasons for the whole party to do what faction B secretly wants it to do.
Ensure that both factions have some goals which simply require the total party strength to accomplish. Set up good reasons to do both sets; make all goals seem like reasonable choices. (For bonus points: then make sure there isn't time to do everything, and enjoy watching both factions fake evidence for why their project is the biggest problem.)
On that theme: mix small and large goals. When faction A has failed to convince the party to attack a wizard's keep, they'll still feel satisfied that they recruited the nobleman's cellarer as a spy. A mix of goals lets everyone feel they can accomplish something every session.
Major-goal-failures don't end the game for either faction. This is important, or your party breaks apart. When the factions' goals for the party are opposed (or there's only time to accomplish one), lower the stakes - make sure the players don't think their goal is a life-or-death-necessity for their subgroup. Make sure the losing conspiracy can think "That hurt... oh well, better luck next time" and go on to the next scheme. You want the goal to seem to seem important... but not enough to kill the rest of the party over, thus losing their resources. (Real power groups rarely bet everything on one plan.)
It seems obvious to establish a note-passing rule for people to scheme secretly, but I advise against it. It brings the intrigue into an obvious, game-slowing mechanic. Instead, be sure to ask the factions secretly between sessions - not only what they plan to do next time (and what excuses they'll use), but also for what they just did. Give the players a limited power to retcon the explanations for their own actions, and they won't need notes, just cover stories.
("When I took that bandit prisoner off to the city guard to be hung, I actually dropped him off with my family's militia for interrogation. Once we know where they hid the loot, our side can send soldiers to go get it.")
Take notes on those cover stories, they can lead to consequences later... making players scramble to make facts fit the story they'd already told the rest of the party.
Despite what's been said in some answers here, I think D&D can be used perfectly well. I've run this approach to good effect in several Star Wars and Werewolf games, which are just as conflict-oriented. The key is to ensure that neither faction's goals are well-served by the total collapse of the party.
Even enemies aren't adversarial all the time. Be sure that both factions have some goals that the other group simply doesn't care about either way. You can create the feeling and tension of intrigue without risking a party feud when tasks fail. (Bob ends up running out of arrows to make a plausible excuse for visiting the markets before storming the overlord's tower; actually he just needs a chance to slip his side a magic statue they gave him to transport.)
Let the factions disagree on methods or outcomes, not primary objectives.
Example: I ran a two-party competitive Star Wars game in which the players were Rebels and Smugglers. Both want to smuggle firearms past the Empire... but subgoals vary. When the stormtroopers pour out, they can fight together as a cohesive party. But if the stormtroopers attack some other random facility... the rebels want to act, and the smugglers want to lay low and get paid.
- Make sure have different aims in the grand scheme of things, but not inherently opposed ones. If the factions are mortal enemies, this can only end in a firefight. You want the game the factions are playing with each other to be non-zero-sum. Some actions should help both, or hurt both.
Example: Factions A and B wanting different candidates to inherit the throne is bad; one's gain is another's loss. Faction A wanting more money for the military and B wanting a balanced budget is better. There will be conflicts, but also some situations will make both factions happy (more farm tax income), and some will make neither happy (loss of expensive equipment). They may scheme to get different people on the throne... but that won't be the point for them, just a means, so the party doesn't have to break apart over it.
- Make sure both factions have plenty of leverage over each other. You want party conflicts solved by roleplay and intrigue, not firefights. So give them things to negotiate with. When faction A doesn't accomplish their goal because the party does something else, the players will feel like they traded off one priority in order to accomplish another objective. This maintains the priority of intrigue and secrecy over just arguing. More importantly, it avoids one faction's players feeling that they were "beaten" or "outplayed" by the rest of the party - they at least feel as if they gained something for their side, even if it wasn't their ideal goal.
Example: Let a mage in faction A have research notes he's not using, that faction B would love to have. Make sure faction B's thief hears information that can prevent (or cause) blackmail attempts against A's priest. Let A have all the rich characters, and B always need money, but have much better contacts.
Major villains should be existential threats to both factions. (This needn't spoil the theme of intrigue; a new player trying to turn one faction into attacking the other is an enemy of both... and both sides may have independent reasons to deal with it.)
Be cautious with your intention to put the experienced players in one faction and the new ones in another; it leaves you without a character present to 'smooth the way' if the newer players struggle with a goal. Consider putting one experienced player in each faction, and letting them 'recruit' the rest of their group in-game.