You first find Vincent in a coffin in the basement of the ShinRa Mansion. Cloud’s and Vincent’s lines remain the same regardless of who you have in your party. Only one other party member will speak to Vincent, but if you try different combinations of characters, you can get everyone to speak in the “Awakening Vincent” dialog. I was unable to get Red XIII to speak, but I've found his dialog on Yinza's website. It is also possible to skip getting Vincent, as he is a “hidden character” and get him after you aquire Cid.
After reading Hojo’s note and following the clues within, Cloud and the party obtain a key to the basement room Vincent is locked in. The party goes down stairs and opens the door. The room has several coffins, one of which lies unopened in the center of the room. Cloud raps on the coffin lid.
Vincent: “….to wake me from the nightmare.”
The coffin lid flies open to the left of the room and hits the ground. Vincent sits upright in the coffin.
Vincent: “Who is it!?”
Vincent: “ …Never seen you before. You must leave.”
Cloud: “You were having a nightmare.”
If:
Aeris: “You’ll dream about bad things if you sleep in a place like this.”
Barret: “I woke ya up. Ya atta be thankin’ me.”
Cait Sith: “Aren’t you glad you woke up?”
Cid: “The air’s pretty heavy here. How can you sleep in a place like this?”
Red XIII: “...You don't look good.”
Tifa: “How bad was your dream?”
Yuffie: “Man, you really WERE having a bad dream! Even I feel bad!”
Vincent: “Hmph…a nightmare? My long sleep has given me time to atone.”
Cloud: “What are you saying?”
Vincent: “I have nothing to say to strangers. Get out. This mansion is the beginning of your nightmare.”
Cloud: “…You can say that again.”
Vincent: “Hmph… what do you know?”
Cloud: “Like you said, this mansion is the beginning of a nightmare. No, it’s not a dream, its for real. Sephiroth has lost his mind. He found the secrets hidden in this mansion…”
Vincent: “Sephiroth!?”
Vincent & Cloud: “You know Sephiroth?”
Vincent does a slow back flip out of coffin and lands on the lid.
Vincent: “You start first.”
Cloud: (does hand gestures that indicate he is explaining the situation) “…..that’s how it was.”
Vincent: “So Sephiroth knows he was created five years ago? And about the Jenova Project?” (I believe “created five years ago” is a mistake in the dialog. Given the game timeline, Sephiroth could not have been created five years prior to present game time. The line should have sounded like this: “So five years ago Sephiroth learned of his creation?” That would make sense with Cloud’s flashbacks to Nibelheim from five years ago, which show Sephiroth learning of his past at that time.)
Cloud: “He was missing but has just recently reappeared. He has taken many lives and is seeking the Promised Land.”
Cloud: “Now it’s your turn.”
Vincent: “Sorry….I cannot speak.”
If:
Aeris: “That’s dirty!”
Barret: “Hey! You lyin’ white-faced!”
Cait Sith: “Hey, that’s not right…”
Cid: “Good! I’m not in the mood to hear long stories. Let’s get the hell outta here!”
Red XIII: “What... you're serious?”
Tifa: “Hmm… I’m disappointed.”
Yuffie: “Oh, that’s okay. It was probably depressing anyway.”
Vincent: “Hearing your stories has added upon me yet another sin. More nightmares shall come to me now, more than I previously had. Now please leave.”
Vincent goes back into the coffin and the coffin lid shuts itself.
Cloud raps on the lid and Vincent comes out again.
Vincent: “You’re still here”
Cloud: “At least tell us you name.”
Vincent: “I was with the Shinra Manufacturing Department in Administrative Research, otherwise known as the Turks….Vincent.”
Cloud: “The Turks!?”
Vincent: “Formerly of the Turks. I have no affiliation with Shinra now. ….And you?”
Cloud: “Cloud, formerly of soldier.”
Vincent: “You were also with Shinra…? Then do you know Lucrecia?”
Cloud: “Who?”
Vincent: “….Lucrecia. The woman who gave birth to Sephiroth.”
Cloud: “…gave birth…? Wasn’t Jenova Sephiroth’s mother?”
Vincent: “That isn’t completely wrong, but just a theory. He was born from a beautiful lady. That lady was Lucrecia. She was an assistant to Professor Gast of the Jenova Project. Beautiful…Lucrecia…”
Cloud: “…A human experiment.”
Vincent: “There was no way to cancel the experiment. I couldn’t stop her. That was my sin. I let the one I loved, the one I respected most, face the worst.”
If:
Aeris: “So your punishment was to sleep? That’s…kinda weird.”
Barret: “That why you started sleepin’ in a box? Gimme a break!”
Cait Sith: “You think that’s punishing yourself? You’re just sleeping.”
Cid: “You shouldn’t sleep just because of that. Sleep?”
Red XIII: “So you slept as punishment? Isn't that strange? That thinking...”
Tifa: “So the punishment was sleep? That’s weird.”
Yuffie: “Ok. Boring story no.1.”
Vincent: “Let me sleep.”
Vincent closes the coffin. Cloud leaves the room and heads for the stairs. Before he can make it to the stairs, Vincent comes out of the room.
Vincent: “Wait! If I go with you will I meet Hojo?”
Cloud: “Dunno. But we’re after him and Sephiroth. So, I guess sooner or later…” (Note: Cloud says something else if you wait until near the end of the game after you’ve already killed Hojo to get Vincent. I don’t recall the exact words because I only saw it once at a friend’s house, but Cloud hesitantly lies to Vincent. He does not reveal that Hojo is already dead.)
Vincent: “Lucrecia….All right. I’ve decided to go with you.”
If:
Aeris: “Sure is a quick change of heart. Huh?”
Barret: “W,what?”
Cait Sith: “What?!”
Cid: “Hey, hold on sleepy. You’re pretty damn pushy!”
Red XIII: “What?”
Tifa: “What? You’re coming with us?”
Yuffie: “Huh? Oh great just what I needed…”
Vincent: “Being a former Turks, I may be of help to you…”
Cloud: “All right, then.”
Vincent joins your party.