Like with so many things, I find having a template that distills the character down to the core things I need to have on hand works well here. And for intrigue we need to know a few things, but none so important as what it is the character is actually aiming for - in the long run - and what levers exists to try and budge or move them.
For that, I've recently come up with four things that I feel will give me a good grasp on these characters. The template, effectively, looks like this:
Self explanatory, this is who and what the character is. What only if it is relevant. For example, just the name "Bayushi Saki" tells me the character is a Scorpion character in Legend of the Five Rings, while in a D&D game I might want to have it be "Bayushi Saki, Dwarven Lord."
Try to keep this succinct. You want one or two physical characteristics, and one or two mental characteristics. Lean on the things that make this character stand out. If red hair is very common in this area of your setting, there's no need to call out all the red-heads. Just like in most Fantasy settings - unless you've made a point to differentiate characters before - you can get away without calling out that all the pretty people are pretty. Most players just assume everyone is at least "could be played by a leading actor/actress" beautiful.
We've talked about need a lot lately here. Suffice to say, this is the 'emotional need' that the character is looking to have fulfilled. The thing at their core that drives them. This is important because even if the players can't help with what the character wants they may be able to budge the character by feeding that need.
What is the character actually out for? This can be as simple as "A good time" or "A worthy enemy" or more specific like "to expose
for the charlatan they are." This is the main 'want' that will drive the character in the game. The method by which they are hoping to achieve their Emotional Need, or to deny and hide that need.
This is an Intrigue Setting. That means all your main players should have at least one secret. This secret should be something that the character believes would destroy them if it got out. Something they are willing to act to protect, to compromise or move on things to keep quiet. Something they could potentially be blackmailed for, or that could drive them to a different course of action. A lever for control, if a very dangerous one. And yes, it could be someone else's secret that they're protecting, as long as it gives the people who know it that lever.
Want & Secrets.
In practice I've found Want & Secrets to be the hardest of the four to do. Mostly as when mass producing NPCs it can be hard to give everyone a practical goal, while I'm perfectly fine just randomly assigning a one word emotional need and seeing how it plays out. Secrets can also be awkward, since you want them to be significant, and clearly significant. Just remember, it only has to be significant to the character.
For example, a younger character with very controlling parents secret might simply be that they're reading books by an author their parents have forbidden to them. Its not the end of the world, but if the character believes it will be bad for them if known, and if the character will act to protect that secret, it is good enough.
Not every secret, or goal, has to be to life changing. It just has to be important to the character.