I ran a political game a few months back that fell apart after four sessions, and wasn't really fun for anyone. I'd have two pieces of advice based on that experience.
1) Make sure that everyone wants to play a political game. One person saying they want it and everyone else staying quiet almost certainly means that the others don't want a political game, but don't want to ruin someone else's fun. This was the problem I ran into.
2) Start small, and let the players determine how their network spreads. Before I tell you how to do that, I'm going to explain how I manage factions and NPCs (who are just really small factions).
A politcal game is all about factions, ranging in size from a single person to alliances of organizations. Their influence will range from someone willing to act for themselves all the way up to leading or controlling thousands of people. A single player character is on the low end of both (at first). The government of a country has both large membership and influence. The king at the head of that government is a single person with immense influence. A band of poorly equipped escaped slaves could be a faction with large membership but poor influence. A single faction will generally be composed of several smaller factions, which can be further divided until you reach individual members.
With that background, in the first session I will introduce roughly five factions to the players.
There will usually be one that is motivated to be hostile to the PCs, but cannot act against them immediately. In my game the PCs were secretly devil worshippers, so I used a large monotheistic clergy that included a large portion of the nation's nobility. This is generally a lurking threat that the players are aware of constantly, but unable to effectively influence at first. It shapes their actions without a direct confrontation.
The second faction acts as a foil to the first, helping the party or acting as a patron. For my party this was the devil that they worked for. I like to use a hands off approach for this group. The devil gave the party a broad task and a lead to follow: have this monotheistic nation accept the worship of other deities so big man Asmodeus can set up churches here. I suggest you use the nobility's annual tournament to ingratiate yourselves with their society.
The third faction is an immediate villain that the party can work against to accomplish their objective. I used a duke, brother to the king, leader of the nation's armies, and reigning champion of the tournament. He hires mercenaries and champions every year to maintain his victory. He was a paladin of the monotheistic clergy, but that wasn't confirmed beyond rumour until they fought him in the melee. Finding a way to make the party oppose this faction is best handled by a (possibly disobedient) subordinate. I had his arrogant champion challenge a PC to a fight and, after winning, taunt the others that they might as well back out of the tournament. My party didn't meet the Duke until the third session, and by then they had all but declared war on him.
The fourth faction is a rival to the immediate villain that the party's immediate goals align with. My party needed to find allies to face the King's brother, so I introduced a faction that hadn't been created yet. I added a half dozen independent nobles, mostly low ranking, who were tired of the Duke's victory every year. The party's first real accomplishment was bringing them all together, though they themselves did not have the political influence to head this new group.
The fifth faction is a neutral party that the party can use to their advantage, but without any pressing loyalties one way or the other. This was the knight in whose village the tournament was taking place. He offered housing for the party as well as an invitation to the tournament. He was also the most likely way that they could have been exposed. My players did not take to this character too much, so he fell offscreen pretty fast.
It looks like a lot, but this is starting small. Some of these factions will only show up once every few sessions, while others will Now that you've got a good sized roster of factions for the players to interact with, just sit back and watch them work. If they are interested in politics they will develop a relationship with some of the factions you've introduced. As time goes on, you can introduce more factions where they make sense. Two of the top members of the factions my players helped create got fleshed out further and became their own sub-factions. The same happened with the opposing faction.
When the party started to feel like things were stable, I'd add a twist and more factions. I like to introduce this shortly before resolving the current issue is resolved so that it feels like there is always something going on in the world. There's no finishing an adventure and waiting around for the next one to show up.
Twist: The noble whose team wins the tournament may claim the right to lead the king's armies (discovered halfway through the tournament). The leader of the party's team is using this to strengthen his domain.
Faction: Rebellious group that wants to overthrow the leader of the party's team and claim his lands.
Faction: Only surviving character from a PC's background is revealed to work for the party's enemy, but is friendly with party.
When my party would latch onto a faction, I considered who else would be present in their sphere of influence, what conflicts and motivations existed between those groups, and then introduced them. By waiting until you know who the party is interested in you can avoid making lots of factions that would end up ignored. At this point you just play all these factions to their motivations and ongoing plots, wrap the PCs up in this endless web, and watch the chaos unfold.
Edit: I use this technique far more gently for NPCs and conflict in normal games. I don't spend as much time developing them in game, and it happens over a longer timescale, but the principle of expanding spheres of influence still applies.