This achievement is story related and unlocks during the French Ward missions. After you secure the French Ward rackets--Sex and Drugs--the
Kill the Judge mission becomes available. Lincoln meets with Donovan and finds out that the Judge rides in a very tough (but not completely impregnable) armored vehicle and that he varies his route.
Whereas the game suggests that Lincoln tail the judge and ambush him, it is much simpler to put a booby trap in the road before the Judge sets out. Open the weapon wheel by pressing
then cycle through the explosives using
until C4 is the current explosive. Lincoln only has to walk a few steps to place a C4 charge on the road near where the Judge will begin his route. Place the C4 charge by pressing
followed by
.
After placing the C4 charge, Lincoln gets in a red De’Leo Traviata that is conveniently parked on the corner. He starts off looking over his shoulder. As soon as the Judge's armored car gets to the explosive, Lincoln detonates the C4 charge by pressing
. Note that the control for detonating the C4 while in a vehicle differs from detonating it while on foot.
If successful, a banner will appear stating, "Kill the Judge--Escape." Lincoln then has to escape the police cordon. The De’Leo can easily outrun the police cars or he can use an associate. When the cordon is cleared a banner will appear stating
Kill the Judge is complete followed by another banner stating that the story mission
Kill “Uncle” Lou Marcano has been unlocked. The achievement will pop after these two banners appear.
The mission can be accomplished in any number of ways. Lincoln can tail the armored vehicle and ambush it using his car, grenades, rocket launcher (if available), etc. I recommend the booby trap approach described above and demonstrated in the below video. Lincoln does not have to deal with the Judge's police protection before or during the assassination and gets a head start escaping the cordon following the kill: