For this trophy, you need to see all of the Diva Room events, including birthdays. These events are triggered with Floor/Shelf Items and Gifts, both of which you unlock in the Rhythm game (mostly) then buy from the shop.
For the Floor/Shelf Item events, you need to place the item from the Remodel menu, then go into the Item Events menu, then simply select the item you placed. Here are the Floor/Shelf Items you'll need to buy along with the songs/conditions that unlock them:
Papercraft Miku - Level 2 Affinity with Miku
Papercraft Rin - Level 2 Affinity with Rin
Papercraft Len - Level 2 Affinity with Len
Papercraft Luka - Level 2 Affinity with Luka
Papercraft Kaito - Level 2 Affinity with Kaito
Papercraft Meiko - Level 2 Affinity with Meiko
Totem Pole - Nyanyanyanyanyanyanya!
Guitar - Odds & Ends
Keyboard - Weekender Girl
Microphone - Black Rock Shooter
Decorative Plant - Negaposi*Continues
Suit of Gold Armor - Hm? Ah, Yes.
Suit of Silver Armor - Unhappy Refrain
Miku Figurine - Level 4 Affinity with Miku
Rin Figurine - Level 4 Affinity with Rin
Len Figurine - Level 4 Affinity with Len
Luka Figurine - Level 4 Affinity with Luka
Kaito Figurine - Level 4 Affinity with Kaito
Meiko Figurine - Level 4 Affinity with Meiko
Eek, a Leek! - God-Tier Tune
SLR Camera - Freely Tomorrow
Handheld Game Console - The MMORPG Addict's Anthem
Cake Assortment - What Do You Mean!?
Tea Set - Black Rock Shooter
Seafood Stew - Continuing Dream
Fortune Box - Sadistic Music Factory
Green Flower Vase - Monochrome Blue Sky
Yellow Flower Vase - Unhappy Refrain
Pink Flower Vase - World's End Umbrella
Fishbowl (Red Fish) - Time Machine
Fishbowl (Black Fish) - Monochrome Blue Sky
Fishbowl (White Fish) - World's End Dance Hall
IMPORTANT!: The Papercrafts and Figurines can only be triggered when in their appropriate rooms. For example, to trigger the Papercraft Miku event, it must be placed in Miku's room. Papercraft Luka in Luka's room, and so on.
ALSO IMPORTANT!: The Fortune Box has 3 different events: Very bad, Just so-so, and Very good. They play at random and they all must be triggered for the trophy. Just keep selecting it until you see all three events.
Now for Gifts, all you have to do is give them to the character. It doesn't matter if they like it or not. Here are the gifts you'll need along with their unlocking songs/conditions:
Orange Juice - Urbandonment
Iced Coffee - Black Rock Shooter
Melon Soda - Secret Police
Taiyaki - Tengaku
Macarons - Hm? Ah, Yes.
Cookies - The MMORPG Addict's Anthem
Donuts - Melancholic
Tuna Sushi - DYE
Rice Omelet - Nostalogic
Birthday Cake (Player) - Level 6 Affinity with everyone
Birthday Cake (Miku) - Level 6 Affinity with Miku
Birthday Cake (Rin & Len) - Level 6 Affinity with Rin or Len
Birthday Cake (Luka) - Level 6 Affinity with Luka
Birthday Cake (Kaito) - Level 6 Affinity with Kaito
Birthday Cake (Meiko) - Level 6 Affinity with Meiko
Sheet Music - Continuing Dream
Hand Fan - Summer Idol
Sunflowers - Time Machine
Tulips - ACUTE
Daisies - Unhappy Refrain
Stuffed Bear - Ashes to Ashes
Stuffed Rabbit - Glasses
Stuffed Cat - Nyanyanyanyanyanyanya!
Paper Fan - Remote Controller
Jack-in-the-Box - Sadistic Music Factory
Maracas - Negaposi*Continues
Tambourine - Cat Food
Calligraphy Paper - Continuing Dream
Sketchbook - Odds & Ends
IMPORTANT!: The Calligraphy Paper has 6 different events. Each event triggers randomly when you give Calligraphy Paper to someone.
ALSO IMPORTANT!: The Sketchbook has 12 different events. Like the Calligraphy Paper, each event is randomly triggered. Here's how it works, though: Basically, the character you give the Sketchbook to will draw a picture of another character. However, they'll never draw a picture of themselves so when you have both events for one character, switch to that one and keep giving Sketchbooks to him/her to get the others. Each character has 2 different events: a good drawing and a bad drawing. This may be difficult to understand, but once you see it, you'll get what I mean.
On a side note, you can also get the Birthday Cake events by visiting the appropriate character's room on their birthday. The dates are as follows:
Hatsune Miku: August 31
Kagamine Rin and Len: December 27
Megurine Luka: January 30
Kaito: February 17
Meiko: November 5
Player (You): Visit anyone's room on whatever date you set on your profile in the Records menu.
There's a quick way to know if you already have an event or not. You'll know if you already have an event if you can skip it. If you can't, then it's a new event. This is especially handy for the Fortune Box, Calligraphy Paper, and Sketchbook events.
Use the Event Collection tab under the menu in Diva Room to keep track of what events you have/need.