To unlock the All the King's Men trophy in Spider-Man, you simply need to complete each hideout on the map marked with a red crown icon. The icon will go grey when you have completed the hideout.
These will appear early on in the game -- you will need to complete the introductory missions until the point where Yuri sends you to one of these hideouts. You'll complete that hideout as part of the mission, at which point other hideouts will be revealed. You'll need to continue further with the story to reveal them all though -- there should be five in total
All of Fisk's hideouts are exterior rooftop locations. You will need to take down several waves of enemies. The first wave can normally be taken out mostly with stealth attacks, after that the enemies will be alerted to your presence. Don't forget that you can always swing a little below the top of the roof for a breather if things get too hot -- just don't leave the area of the mission. Enemies will have standard shields, pistols, stun batons and assault rifles. In later waves you will get some enemies with rocket launchers -- make sure to take these guys down promptly.
Hideouts are vastly easier once you have unlocked some abilities and gadgets, particularly the ability to slow time with a perfect dodge and any ability or gadget that boosts focus refills. Remember -- stay calm and don't button-mash, wait for the enemies to attack and counter. Use height to your advantage but always always be on the lookout for your Spidey-sense and dodge when the aura goes from grey to blue (or an incoming bullet track turns from white to red).
These are the easiest Hideouts to complete so you shouldn't have too much trouble. Don't forget to press R3 when you are in there to remind you which bonus objectives you can complete -- it's not strictly necessary for a trophy, but acquiring additional Base Tokens through completing objectives will help you unlock Suits and gadgets.
Financial District
Bonus Objectives Perform 10 Perfect Dodges
Stealth Takedown 3 Snipers
Rewards 3 Base Tokens
Greenwich
Bonus Objectives Perform 10 Finishers
Web 5 Enemies to walls
Rewards 3 Base Tokens
Upper West Side
Bonus Objectives Perform 3 Stealth Takedowns
Throw 10 objects
Rewards 3 Base Tokens
Upper East Side
Bonus Objectives Perform 3 Perch Takedowns
Knock or throw 5 enemies off the building
Rewards 3 Base Tokens
Harlem
Bonus Objectives Disarm 5 enemies
Perform 10 Air Launch attacks
Rewards 3 Base Tokens