Deathmatch

This is a simple mode that requires only 2 people. You will more than likely already have finished one match of this before you start going for Full Monty.

Team Deathmatch

From my understanding, this game mode is functionally identical to large team deathmatch. Be sure to choose team deathmatch and not last team standing. Also only requires 2 people to start.

Payne Killer

This is a mode from the deathmatch made in heaven dlc, and it requires 3 players to start a match. Progress for Full Monty can easily be paired with the "Long Arm Of The Lawless" achievement, by setting your deathmatch avatars to DeMarcos Abroad and picking the canal de panama map. Be sure not to pick a map that says "Saci is loose" before the description, as this mode is not required.

Gang Wars

This includes 12 modes all but three of which are randomly generated. If you are missing delivery, the map for it is the "It Goes Round and Round" version of the tiete river docks map. The other two modes that are not random are Survivor and Showdown. These two modes can be found only at the end of the gang wars match. All of the gang war modes are listed below.

Turf Grab
Passage
Total Turf
Last Man
Warfare
Siege
Grab
Delivery
Short Fuse
Takedown
Survivor
Showdown

In order for the mode to count toward completion you must have stayed until the end of the match, either returning to lobby or moving to the next gang war mode. To keep track of which modes you have completed in gang wars, you can screenshot the menu just before the last mode that shows how many points each team gets for winning previous modes.

Credit to ZombieMaze for list of modes.