After you complete the main storyline quest
Home Sweet Home you will unlock your 4th weapon slot, which will allow you to earn this trophy. You cannot earn it before that point.
You can obtain purple or orange weapons through regular gameplay, whether its from random enemy drops, loot/weapon crates, weapon crates that require moonstones, quest rewards, or using golden keys via SHiFT codes.
If you want to avoid Golden Keys, then simply play the game normally and save a purple or orange item in your vault if you plan on switching it out for a lower rarity weapon when your level exceeds its use. Once you have an item for each slot, equip it and your trophy will pop.
The slots you must fill are:
-All 4 weapon slots
-Grenade Mod
-Class Mod
-Oz Mod
-Shield
If you want to fast track this process, simply enter some SHiFT codes for Golden Keys, open the chest in Concordia, and collect purple level weapons with every single chest opening.
Below is a list of SHiFT codes available on Orcz.com
Make sure the code is still available and hasn't expired yet
http://orcz.com/Borderlands_Pre-Sequel:_Shift_Codes