You don't get an army by adventuring - you get an army by establishing yourself with a domain, and then using that domain to form a realm. When you have a domain, your army is paid for by the families (and it's unavoidably so, due to the garrison expense rules). So even characters who want to maximize domain profits will still have an army of around 2gp/month/family in size.
The more challenging question is "what sort of campaign world do you need to set in motion to create opportunities for player characters to become conquerors and kings"? The solution I have found is to have a situation with a conventional and existential threat. The conventional threat is a political threat that distracts the existing realm and domain rulers. The existential threat is a campaign-level danger that slowly grows more threatening over time, and is linked to the party's various dungeon escapades.
The campaign thus follows this stage:
Adventurer - NPC rulers are busy playing the game of thrones, and delegate adventuring to PCs. The PCs, while exploring dungeons, discover hints of dark forces at work.
Conqueror - Some casualties have occurred among the NPC rulers, due to their conventional struggle, resulting in opportunities for junior rulership slots for PCs. The dark forces are becoming more dangerous, but as the stakes of the conventional struggle are now so high, the main NPC rulers are still ignoring them. This gives the more justification that more sentimental contemporary adventurers need to justify seizing power. (I've found many players are so ingrained in contemporary values of democracy and humanism that they just can't bring themselves to crush their enemies and seize the thrones with no justification save power.)
King - The PC rulers unify the region by defeating any remaining conventional foes, then turn their realm-level resources towards dealing with the existential threat, which has now become severe.
In the two major ACKS campaigns I've run, the players have ended up as rulers of a principality-sized domain by about 10th or 11th level. That's about 50,000 to 100,000 families, yielding 100,000 to 200,000gp per month for an army.
In the Auran Empire campaign, the adventurers were facing (a) a borderlands left in chaos by the withdrawal of the Auran troops and (b) the Awakening of the last sorcerer-king of Zahar. They were eventually elected military-commanders of the borderlands, and led the local legates to defeat the last sorcerer-king of Zahar (though at terrible cost). They then had one of their member seize rulership of Aura as the "Theocrat" (14th level cleric) and fought some pitched battles against other exarchs vying for the throne. They weren't able to conquer the whole Empire but did secure the place of the borderlands (cutting away chunks of S. Argolle and Krysea) as its own exarchate.
In the Opelenean Nights campaign, the adventureres were facing (a) an uprising from native Opeleneans against the empire and (b) a Thrassian tyrant waiting in torpor in a tomb. Whilst adventuring they accidentally woke up the Thrassian tyrant and got a town destroyed, which led to them be exiled into the Waste. There they joined the uprising and became its leaders, and then justified the takeover of Opelenea as necessary to unit against the Thrassian threat. Then they lost and all got killed to the Thrassian tyrant.
Hope that helps!