This game is a really fun experience,
Achievement integration and picking the right milestones as achievements are quite well done too,
So naturally under these circumstances, I went for finishing the game 100 percent.
And as a perfectionist strategist, with kind of bad reflexes, at some point I was quite puzzled on how I could ever finish a run in 10 minutes.
After some playing around all the classes to find the right one for the job, at last I finished a run on 9:30 minutes mark, with a full on adrenaline rush, and I decided to share my findings on my journey, to maybe make it easier for the future players.

If you have not done it yet, let's start with getting a "Visible Run Timer" on screen with us.
Go to "Settings" on main menu,
Hit "Display" there,
Then switch "Visible Run Timer" on.

Count Leo with his Shadow class is the man you wanna hire for a very quick dethroning mission.

There are two main reasons for this:
1. While every character has very different ways to gain shop currency, Leo's (within Cymbidium category) "Shrouded" skill proves to be best: Because sending patrols home and exploring tiles takes a huge time we don't posses, so you will want to minimize all that, yes even challenging patrols for gold needs to be minimized. And with that gaining "+1 shop currency for every 4 hidden tiles when finishing a level" really pops.
2. Shadow's legendary "Backstab" skill, which has a whooping 2 damage, quickens the boss fight with the King, and within the levels if you really must fight a patrol for some quick currency, shortens the time needed for sending a fella back home. And on top of it all very cheap as well.

And a third bonus reson:
3. Shadow's energy refilling skill "Prepared", saves us lots of time on planning and calculations, because as your energy resets every level, you can just focus on that level. For other classes, your energy levels for the current level can affect the next ones, so more often than not they require some rationing of energy between levels, and that rationing will take from your precious time.

Side note here: Of course you can pause the game whenever you want to make the calculations and rationing on free time, but I have found this strategy is taxing on your flow and on your command of the tiles. Still a viable strategem nonetheless.

Ohoho, you are going to step on a lot of patrols on the road, so you better prioritize getting health from shop.
Refilling your smokebomb is the next priority, if there are 2 or 3 enemies next to each other, it is perfect for smokebomb, don't even think about it. For single enemies just use your "Cartwheel" to hide them efficiently.
With the rest of your shop currency, you can spend however you want. Usually any item, is better than not having one at all.

Now from our beloved wise Smith, the skills you want to get, there are couple of really good ones:
1. "Dodge: you take no damage from stepping on a hidden patrol": You can count this skill as another reason why we went with Leo. With this now, without even blinking for a second, you can just franticly run around on all the tiles whatsoever, saving huge time. The "Cartwheel" really shines when coupled with this skill here.
2. "Magic Flute: hide one random patrol 1 time per level": Really powerful on its own, in all gameplay sessions. Always useful, especially for the ones already visible from the beginning of the level.
3. "Surprise: cause +1damage when attacking a hidden patrol": on normal difficulty, even if you don't get any other damage options from shops, you can just dethrone the king in 2 "backstabs".

Now, altough these are extra useful, you can also manage to finish in 10 minute mark without getting any of them as well. No reason to roll for new ones until getting these, or if you don't get them just ending a good run there. This challenge is not much about the right skills to get in-game, but rather building the skill of having nerves of steel as a player under all circumstances.

In the gym feel free to upgrade anything, everything is going to be equally useful in their upgraded form.

I believe in you, have fun out there bringing justice to once untouchable kings.