I would assume its based on the tradition of St. Peter basically being Heaven's doorkeeper, which is a folk belief that comes from when Peter affirms that Jesus is the Messiah and Son of God, and Jesus says:
"And I also say unto thee, that thou art Peter, and upon this rock I will build my church; and the gates of Hades shall not prevail against it. I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven; and whatsoever thou shalt loose on earth shall be loosed in heaven."
The thing I find weird about Peter in the show is that he seems just like some preppy dork. St. Peter was a blue collar fisherman, and at one point almost got himself killed when according to the Gospel of John, while a bunch of soldiers are trying to arrest Jesus, "Simon Peter, who had a sword, drew it and struck the high priest’s servant, cutting off his right ear." Yet the characterization Viv got out of this is apparently Kenneth from 30 Rock, without the bizarre half-hinted at backstory.