Welcome to a grindfest and a half.
For this trophy you will need to get to level 72 which is around 41,760,000.
Now to put it into facts I was around 10 million xp when I got my last trophy in the game (500 kill assists). This easily is a 100+ hour trophy (and for a silver, may turn away many) and as the only way to rank up is by playing online, you will encounter the rare occasional random.
However this trophy (like all the others) can be obtained versus bots in public matches.
The best way to get xp is by searching in custom match:
Map - Factory
Game Type - Extraction
Rounds - 13
Time Limit - 2:00 (this really does not matter and you can switch between the other time limits to avoid any other people online.
Friendly Fire - Off (Doesn't matter but as the ai is horrid it will most likely cause them to kill each other from time to time)
Auto Map Cycle - Off (You want this off so it will bring you to the menu each time)
Number of Players - 8 vs 8
CPU - Allowed
Now join the Mercenaries side. When you spawn run to the staircase to your right and crouch on the bridge shooting all the enemies. Your goal is to kill all the enemies to get a 6000xp point bonus (its called ace). It is very time consuming, and VERY boring, so unless you want to finish the game, this really is you best bet, unless you want to play the other modes (which are dead so it will be you against the bots until the occasional person crashes in.) You will average around 12000xp this way and is infact the fastest way:
XP breakdown: (target of what a round should be)
Kills: 4000 (500 each)
Kill Bonuses: 4000 + ? (This will vary by how the enemies are killed but you will always have atleast 4000 from here for all 8 kills)
Victory Bonus: 1500xp
(Note the following solution is based off this vid:
so thank you Stelthy for bringing this to our attention)
There is also another method that I use while not at the console.
I mentioned that there is a victory bonus that gives you 1500xp. Well infact you can get this really by means of idling. If you set a custom game on Factory - Extraction and change the auto map cycle to on, matches will occur between any bots in the match making xp obtainable to you, however some maps are made to have other team, have advantages making you not get a win because the other team has the best postition (7 v 8)
The catch is that you only get xp for winning, if you lose it doesn't count to your total. You also will get around 175k for 8 hours of idling making it a fairly unrewarding reason to leave your system on, (it could be fixed by playing the first method for around 30 mins) However as the grind is boring I myself am balancing them both.
Try to have fun with it, this requires alot of dedication, and the fact that it will be you versus endless numbers of horrible ai, will make it frustrating all I can say is good luck.