This achievement is for completing a Seasonal Manhunt on challenging global difficulty or above.
Seasonal Manhunts are unlocked when you complete the Warlords of New York story. The current season at the time of writing this is season 4, and there are about 16 weeks left. The information here should be very similar to other seasons though, as I believe they mostly follow the same structure.
There are two ways to go about this. The first way, as others have mentioned, is to level up to level 48, which will currently unlock the season 3 seasonal manhunt. Completing that on challenging difficulty should unlock the achievement.
The second way is to use the current season manhunt target, which I haven't been able to find a good explanation about, but that's what I'm going to do here if you choose to go with this route (which I believe is the shorter, less boring one). You can find the information by hitting the select button, and navigating to the manhunt tab. Here, you'll see 5 targets: 4 minor ones and the main season target. For season 4, they are as follows:
Minor Targets:
Viper
Magnus
Kestrel
Circe
Main Target:
Faye Lau
Each target has a series of subtasks you need to complete before you can eliminate them. You need to eliminate all the minor targets before getting the mission to take down the main target. These consist of taking over all control points in the area, eliminating the manhunt specific bounty (when choosing a bounty, choose the one with the manhunt logo beside their name, as there are other bounties as well that won't count), and completing missions. Below are the subtasks you need to complete to unlock the bounty for each target, and each area you need to do it in:
Viper:
West Potomac Park:
-Take over 3 Control Points
-Complete 1 Manhunt Bounty
-Complete Lincoln Memorial on Normal difficulty or above
East Mall:
-Take over 4 Control Points
-Complete 2 Manhunt Bounties
-Complete American History Museum on Normal difficulty or above
Southwest:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete Jefferson Plaza on Normal difficulty or above
-Complete Space Administration HQ on Normal difficulty or above
The Pentagon:
-Complete The Pentagon on Normal difficulty or above
-Complete DARPA Research Labs on Normal difficulty or above
Magnus:
Downtown West:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete Bank Headquarters on Normal difficulty or above
Constitution Hall:
-Take over 3 Control Points
-Complete 1 Manhunt Bounty
-Complete DCD Headquarters on Normal difficulty or above
West End:
-Take over 3 Control Points
-Complete 1 Manhunt Bounty
-Complete Federal Emergency Bunker on Normal difficulty or above
Roosevelt Island:
-Complete Roosevelt Island on Normal difficulty or above
Kestrel:
Downtown East:
-Take over 4 Control Points
-Complete 1 Manhunt Bounty
-Complete Grand Washington Hotel on Normal difficulty or above
Federal Triangle:
-Take over 1 Control Point
-Complete 1 Manhunt Bounty
-Complete Jefferson Trade Center on Normal difficulty or above
-Complete ViewPoint Museum on Normal difficulty or above
Judiciary Square:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete District Union Arena on Normal difficulty or above
Circe:
The Summit:
-Complete a Boss Floor
Battery Park:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete Pathway Park on Normal difficulty or above
Financial District:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete Wall Street on Normal difficulty or above
Civic Center:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete The Tombs on Normal difficulty or above
Financial District:
-Take over 2 Control Points
-Complete 1 Manhunt Bounty
-Complete Stranded Tanker on Normal difficulty or above
Once you've done each target's tasks, you'll unlock a bounty to kill them. Once you've killed all of them, you'll unlock Faye Lau's mission, which is Camp White Oak Manhunt: Faye Lau.
Before starting this mission, go to the map and hit right on the D-Pad. Change the global difficulty to challenging, then hit X to confirm the settings. The game will reload after about 10 seconds. Once this is done, double check the difficulty of the mission by hovering over it and holding Y, then choosing Difficulty and ensure it is sent to Challenging. If your global difficulty is set to Challenging, the mission should be set to Challenging by default, as you cannot choose a difficulty lower than the global difficulty you're currently at. Essentially, as long as the mission difficulty is set to Challenging, it shouldn't matter what the global difficulty is. Once you've confirmed that, hit Y and A to matchmake, or you can try to do it solo. Simply complete the mission and you'll unlock the achievement.
I was season level 44 when I unlocked this, so I felt this to be a bit faster and less repetitive than simply grinding to level 48 to unlock a previous one. At the end of the day though, it comes down to preference as to how you want to approach this, and hopefully this gives people looking to start the DLC a good explanation of the requirements for the achievement.
Also, I played on normal global difficulty for all of the missions and tasks except for the final mission, so you can do everything pretty easily on normal difficulty solo up until the final mission.