Customization items are unlocked throughout the story and VR missions. In order to acquire all customization items, you will need to obtain nearly every collectable (except saving people and the humanoid gekkos) and complete every VR mission with the top rank.
DLC weapons do not count - the only items that will unlock this trophy are the items you find in the game.
Also, you do not need to upgrade every item completely - simply purchasing every item in the customize screen will unlock this trophy.
Let's start with the easy unlockables (in order you will naturally receive them):
Custom Body: Unlocks automatically at the start of R-01
Pole-Arm (L'Étranger): Complete File R-01
Mariachi Uniform: Complete File R-02
Tactical Sai (Dystopia): Complete File R-03
Pincer Blades (Bloodlust): Complete File R-04
Custom Body (Desperado Ver.): Complete File R-04
HF Muramasa Blade: Complete the game on any difficulty
Fuel Cells (x5): All become available by the end of the game
Endurance Cells (x4): Same as Fuel Cells
Skills (x14): All become available as you unlock skills and secondary weapons.
As you can see, you unlock quite a bit just playing the actual game. So let's move on to the not-so-easy unlockables, starting with the HF blades:
High-Frequency Machete: Acquire 10 storage devices
Stun Blade: Acquire all 20 storage devices
High-Frequency Wooden Sword: Find all Men in Boxes
Armor Breaker: Acquire all 30 left arms
High-Frequency Long Sword: Complete, and Rank 1st, in 100% (20) VR missions
Once you meet the requirements for the item, finish the mission. The weapon (and trophy if applicable) will unlock at the results screen.
Now, for the different cyborg bodies:
Custom Cyborg Body (Red): Complete 30% (~6) VR missions
Custom Cyborg Body (Blue): Complete 50% (10) VR missions
Custom Cyborg Body (Yellow): Complete 100% (20) VR missions
Standard Body: Complete the game on Hard difficulty
Suit: Complete File R-00 on Very Hard difficulty
Lastly, the wigs:
Infinite Wig A: Complete the game on at least Normal Difficulty and collect 10 left arms.
Infinite Wig B: Complete the game on at least Normal Difficulty and collect 20 left arms.
Blade Mode Wig: Complete the game on at least Hard Difficulty and collect all left arms.
Questions, corrections and neg vote justifications below.