Completed the mission "Long Way To The Top"
This trophy is for completing the final story mission in the DLC, as such it cannot be missed. All this mission entails is defeating the #1 competitor on the leaderboard in the Torgue Arena (name is omitted to avoid spoilers)
Completed all Campaign of Carnage side missions
There are 20 side missions in the Campaign of Carnage DLC. All you have to do is complete them. Some can be done multiple times so don't worry about that. Also once you beat the main story missions you can go back and complete other side missions which you missed. Just check your Mission Log and go to the location it says to find the quest.
In the spoilers below will be all the side missions and where you can obtain them.
Badass Crater of Badassitude
Badass Crater Bar
Collected 10 pictures of Moxxi in Campaign of Carnage
The Portraits of Moxxi are a fairly rare drop from any enemy in the DLC. You simply need to pick up 10 of them. The best place to get this is in Pyro Pete's Bar on the quest Tier 3 Rematch: Bar Room Blitz, as this quest can be done as many times as you want. The best enemies to get this off of are Badass Enforcer's (Normal mode), or Super Badass Enforcers (TVHM).