For a game that recommends wanton destruction to solve every problem, it has a lot of ways to punish you for doing so. This achievement is one of those instances.
There are a total of 55 people of interest to each faction, called High Value Targets -- usually 10 HVTs for each of the 5 factions, but 3 are story related and mandatory -- that must be verified over the course of the game. Verifying an HVT can be done one of two ways: kill the target and photograph their corpse, or melee them with
to restrain them while they are down, then call for a chopper to whisk them away. The former option is a lot easier and a lot more fun because you can call in your biggest goddamn bomb and sift through the bodies later. The latter option is what you are required to do for this achievement. Any time you are tasked with "Capture or subdue the HVT" on the right side of your screen, you must take the target alive.
You can be as surgical as a SEAL, but there is often way too much going on to handle that will ultimately screw up your captures. Most often, the HVT will kill themself if they start high up (and simply fall to their death) or possess an explosive weapon . Sometimes their buddies will get really eager to kill you and toss grenades, fire RPGs, tank shells, helicopter rockets, or anything else at you that will collateral your HVT. In other instances, the HVT is too close to a faction boundary and will be killed by a rival faction before you can sort out the situation. To make a long story short: everything is working against you.
The most glaring problem with this achievement is that there is no stat informing you of how many HVTs you have killed and how many you have captured, so this makes relying on the autosave a very bad idea. While it does not save over your manual savefiles, the process by which you select the unbotched game is convoluted a great deal by navigating the Options and Save Games menus, which can be avoided entirely by just hitting Continue on the main menu with autosave disabled. Consider in addition that in the midst of the clusterfuck this game often becomes when completing objectives that it becomes nigh impossible to keep track of when the game is autosaving, and because of the missing stat to track HVT captures you are literally tossing caution to the wind for no good reason. While playing with the autosave disabled, make a manual save after every successful capture. Do not get overconfident and think you can get 4-5 HVTs in a row, since your hubris will be the cause of two hours' worth of repeated play.
When capturing HVTs, all that matters is that the "HVT Captured" message appears on the screen (or in the log on the PDA if you happened to miss it). It does not matter if the helicopter that "carries" the HVT gets shot down, because the HVT disappears as soon as you load them anyway. Nearly all of my Chinese and AN HVTs were loaded into helicopters that were blown up shortly thereafter (some didn't even leave the ground), so keep this in mind for some of those particularly problematic HVTs.
You want to endeavor to completely finish one faction before moving onto another, including items not pertinent to this achievement like outposts. The only exception is the endgame with the AN and China factions; you must not participate in the Caracas battle until you complete both factions. Choose one of the factions to do first, complete everything they offer, then advance the story missions until it offers you a mission to battle in Caracas and receive the nuclear bomb. At this point, bribe the faction you did not choose and complete all of their missions. It would be easiest at this point to simply finish the game with the faction you just bribed to avoid having to pay a second bribe to the other, but if you really have some moral objection to allying with one over the other then there's nothing preventing you from joining the other guys again. Keep in mind at any rate that they are identical factions with equivalent equipment and both reward you with what you need at the end of the Caracas battle, so it's all on you if an imaginary alliance in a video game is somehow bothersome because of the flag they have.
You will not unlock this achievement until after you verify Solano at the end of the game. The helpful part is that he cannot die and you automatically capture him after a lame quick-time battle if you captured all of the HVTs before that point. If your mercenary pops Solano with their pistol, then something has gone horribly wrong and you will not receive this achievement.