OPERATIONAL & INTEL TROPHIES:
Basic Training 
Complete the Training Mission.
In order for your training to be considered complete, you must complete the initial "escape scenario" at Graybox, and complete all three training sessions; Technical Ops Orientation, Marksmanship Orientation, and Covert Ops Orientation.
Alpha Protocol 
Complete Operation Desert Spear.
Saudi Arabia completed. This trophy will pop during the score screen at the end of the mission "Intercept Shaheed and Recover Missiles"
Operation Blood Feud
Complete Operation Blood Feud.
Moscow completed. Depending on your actions with Brayko, this trophy will pop during the score screen at the end of either the "Assault Brayko's Mansion" mission or "Prevent Surkov's Escape" mission.
Operation Deus Vult
Complete Operation Deus Vult.
Rome completed. This trophy will pop during the score screen at the end of the mission "Intercept Marburg at the Museum of Art"
Operation True Heirs
Complete Operation True Heirs.
Taipei completed. This trophy will pop during the score screen at the end of the mission "Stop Omen Deng at Memorial Rally"
Full Circle 
Complete Alpha Protocol.
Alpha Protocol completed. This trophy will pop during the final cinematic at the end of the mission "Infiltrate Alpha Protocol". It is worth mentioning that this trophy will be awarded regardless of what ending you get, or who survives.
Evolution of an Action Hero
Complete Alpha Protocol using the Recruit background.
When starting a new game, instead of choosing a background such as "Soldier" or "Field Agent" that gives you free ranks in various skills, you also have the option of choosing the "Recruit" background, which starts you off with
zero skills across the board. For best results, this trophy should be earned on your Easy playthrough, to mitigate the additional difficulty of "Recruit" Thorton's complete ineptitude. Earning this trophy will unlock the "Veteran" background which gives you three ranks in all skills at the start of the game, so this should be a priority before attempting the
Hardcore trophy.
Hardcore
Complete Alpha Protocol on Hard difficulty setting.
The Hard difficulty setting isn't significantly more challenging than Easy, assuming that your character favors a Stealth build. The obvious exception is in the hacking and lockpicking mini-games. The time limits are prohibitively short, leaving zero room for error. Also, the games themselves are harder versions of those you'd find on lower difficulty settings. Locks have five tumblers, and have very little margin of error to engage the tumblers. Alarm panels and door locks have 10-12 circuits and more complex patterns, usually causing you to waste time just trying to find the next number, let alone figure out which circuit to close. Finally, the hacking uses smaller code segments, less than half the size of those you'd find on Easy difficulty. In the time it takes you to move a segment from the top of the screen to the bottom, they will have changed position, meaning that you have very little time to spot the target string before you need to maneuver and lock in the segments. You will find yourself carrying a lot of EMP grenades in later missions, often filling multiple inventory slots just so you don't have to hack/pick anything. Usually the cost of buying grenades will be substantially cheaper than the money you'd miss by passing up a safe with a ridiculous lock.
Combat-wise, enemies will take more damage, and you will need to aim exclusively for headshots, however a Stealth/Pistol build renders most of the difficulty increase moot. The combination of Shadow Operative and Chain Shot can get you out of most situations, and defeat most bosses in one or two uses of the skill. Once you reach Shadow Operative (Master), you become a mass murder machine that would make Steven Heck blanche. Supplement that skill with Advanced Stealth Armor and cooldown enhancing perks like "Under the Radar" or "Sororicide", and nothing in the game can touch you. Using this build for my Hardcore playthrough, by far the most challenging fight in the game was when you fought the two random mooks hand to hand in Marburg's study. Leland's got nothing on Generic Heavy #2.
It should go without saying that you will want to choose the "Veteran" background you unlock with the
Evolution of an Action Hero trophy (that is unless you wanted to go the Destructoid route and be stupid). You can remove ranks in skills you don't normally use, and reallocate them to Stealth and your favored weapons to give yourself an even bigger edge right out of the gate.
Ready For Anything
Acquire the vast majority of Intel available in the game.
The description is misleading. This trophy is awarded for purchasing 75% of the total mission intel available from the Clearinghouse. The problem is that the availability of mission intel varies depending on your actions and reputation with certain characters throughout the game, potentially making this trophy impossible to get on certain playthroughs.
By choosing to "Arrest" Nasri rather than "Execute" or "Extort" him, several additional pieces of intel will be available in Saudi Arabia. Having SIE or Albatross as an ally and completing all previous missions will unlock additional intel for the Embassy raid. Befriending Heck before dealing with Brayko will yield unusual results. Sparing Brayko or Rescuing Surkov will yield additional intel for the "Investigate Delivery at Warehouse" mission. If you allied with SIE in Moscow, she happliy provides intel on Marburg for the "Intercept Marburg at the Museum of Art" mission. High reputation with Hong Shi, Steven Heck and Albatross can provide additional intel for several Taipei missions as well. To let you know if you're getting close, at 50% you will unlock the "Be Prepared" perk.