Keep it to the broad strokes. Don't get too detailed. If you have a city, then determine the makeup of it's population normally using the guidelines in the DMG. Use generic statblocks if you need to, except potentially for NPCs above 1st-level (depends on how high level the PCs are, they may not even care).
In any case, have the income of the kingdom be about 10 gold pieces per day per 100 citizens. This is a pretty low tax rate, though, so feel free to increase it to 20 or 30 gold. Tax rates of 25 and higher, though, will strain your peasants and, unless you're at war and able to stoke the fires of patriotism, you'll run into issues with unrest. Note that this is just based on how much the peasantry produces. Taxes on merchants and the like don't really come into the equation, because they would be used to maintain infrastructure, bureaucracy, etc.
As for your military, they are professional, hired soldiers. Followers you have from Leadership play a different role than your actual armed forces. Paying for their upkeep is generally 1gp per day for a 1st-level soldier, not including their equipment, barracks, maintenance of your infrastructure, etc. For soldiers higher than first level, I'd go with (level)^1.5 gold pieces per day for soldiers of up to 5th level or so. Higher-level soldiers are high enough level to be notable NPCs, and their salary should be determined on a case-by-case basis.
Generally speaking, pirating your tax money for personal expenses isn't that wise. You'll exhaust your funding for soldiers and upset the nobles.