Getting all of the trophies/achievements in the main portion of Borderlands 2 will require you to do two playthroughs of the main campaign. On your first playthrough, you should worry on finishing all of the side missions, as well as the main campaign. It is preferable to use either Zer0 or Salvador, as their class-specific trophies/achievements will require them to level up significantly, with Zer0 needing at least level 31 and Salvador needing at least level 21 (level 25 is recommended). You will also want to work on trying to do all of the level 1 Challenges as you play.
On your second playthrough with a particular character, you will want to concentrate on just going through the main story again and getting to level 50. If you didn't finish the Challenge trophy/achievement, then continue working on that to completion in TVHM. You can also take this opportunity to clean up any of the specific trophies/achievements in the game, although it may be easier to do some in normal mode on another character. If you play solo, then it's a good idea to find a friend or get someone online (they need to be on your friends list, though) to revive them while they are in Fight For Your Life.