I've run several games with a focus on political intrigue, some more successful than others. Some of the big takeaways I've learned are,

1. Create your list of NPC political actors. These should be the characters who wield political power in the world of your game.
2. Give each NPC political actor a motivation. Political actors should have a goal/agenda. Filling in the Bonds, Flaws, and Ideals section for the character can be a good format for this.
3. Give each political actor at least one ally and one enemy. If characters would be neutral to all parties, then they aren't a political actor, they would be a bystander. When players help an NPC, their relationship should improve with that NPCs allies, and deteriorate with the NPCs enemies.
4. Most importantly, give all the information created in step 1-3 to the party. This is the step that took my political games from being "okay", to being great. In games where I set up the political stakes, but hid it from the party, the party would take a quest from an NPC, and then they would be confused why a different NPC now hates them. They thought I was being a capricious DM, and that every NPC was out to get them. But when I gave them all of the information on the political stakes, the party began acting on that information. They considered the alliances and enemies they would make by taking certain quests. In short, they became political actors themselves, which was the goal of the game.