In order to get Vincent you will need to open the safe in the Shinra Mansion in Nibelheim. The combination is 36 right, 10 left, 59 right, 97 right. After opening it, you will have to fight the Missing Number boss. Afterwards, you get the Odin materia, the Cosmo Memory item that teaches Red XIII his final limit break, and a key to a locked room in the basement.
Go into the room in the basement and approach the coffin. Vincent is inside. When talking to him, talk about Sephiroth when prompted. Talk to him again and this time ask who he is when prompted. Once you are done talking, leave the area and he will join you.