Requirement: 4 legenday parts for a single character, crystals and artifacts does not matter.
Legendary equipment is recognizable for it's yellow color on the gear customization menu on your character.
This is a time consuming achievement, you'll encounter different different grade of equipment through missions, vendors and boxes, depending on your selected character you can get equipement for them on the iconic missions, it will make things faster to go straight for the multiplayer section, but i'd recommend playing the single player campaign first.
As i imagine you'll be going for the rest of the achievements this will come naturally so you shouldn't worry about too much. To level up your Battle power you'll need to keep equiping your highest gear, burning your lowest gear and getting more gear from boxes, the highest your gear, the better gear you find in boxes.
To get this achievement as fast as you can, just save your legendary equipment (don't let it equipped) and keep burning everything else, once you've found all the required parts (they changed according to the character), equip them and done, achievement unlocked, you can burn your lowest pieces and keep upgrading your set.
Edit: As CosmicRunaway pointed out, make sure to exit the menu after equiping all the parts, standing on the customization menu doesn't work.