Vincent Character Unlock Side Quest - FF7
This quest takes place inside of the Shinra Mansion which is found in Nibelheim and can be done when you visit Nibelheim for the first time shortly after completing Cosmo Canyon. Technically the main reward for this quest is unlocking Vincent as a character but you'll also get the Odin Materia and Cosmo Memory, Red XIII's ultimate Limit Break.
For this quest the first thing you need to do is defeat the Lost Number boss which is found by opening up the safe in the Shinra Mansion. In the entry room of the mansion go up the stairs and into the west room, this is where the safe is.
In order to open the safe you will need to input the correct combination, the combination is listed below but if you want to figure it out yourself use the piece of paper in the left room by the dungeon's entrance. Note: Before attempting this boss it is highly recommend you save your game. This is by far the most difficult boss you've faced thus far.
This bosses main quirk is when you get him to 50% he will transform into either a fully red or fully purple boss. It all depends on what type of attack drops his health below 50%, if it's a magical attack he will transform into his magical form and if it's a physical attack he will take his physical form.
I'd recommend trying to get him to change into his magical form as that is easier to deal with - his physical form has an attack that can hit for 2k which is no good. If you took my advice earlier in the walkthrough and got the Big Guard Enemy Skill, this boss is an extremely good time to make use of it.
After the fight is over grab the Key to the Basement from the safe and the Odin Materia on the ground nearby. Our next destination is back down in the basement in the room next to the library that Sephiroth went insane in. You'll find a purple casket inside of the room, inspect it to trigger a scene with Vincent.
When Vincent tries to go back to sleep say to him, "Oh well, I was only going to talk about that Sephiroth guy" and he'll be all like, "NANI?!?!" and ask you to tell him your story. He declines to tell his side of the story and goes back to sleep - F that.... Interrupt him again and ask him "Who are you?" for more answers. After this try to leave the area and Vincent will stop you and join your party.