The phrase “personal relationship with Jesus” is not to be found in the bible, to answer your direct question.
Many have stated that the idea behind the phrase can be found throughout the New Testament, which is true in the sense that anyone who accepts Christ enters into a relationship with Jesus that is real and personal. Howevever, there is a catch.
In evangelical churches, where the phrase is tossed about often and enthusiastically, “personal relationship” means several things that go beyond the plain meaning of the words.
First, a “personal relationship” is contrasted with the “impersonal relationship” that you have by simply joining a church and participating in its religious services, especially if that church is liturgical. How could you possibly have a personal relationship if that relationship is mediated by priests and ceremonies? A personal relationship requires interacting directly with God – cut out the clerical middle-men! So, you don’t use standard prayers, you don’t pray to saints, you don’t confess sins to a priest – you go directly to Christ, just as you are, with your own words. Your interaction with God must be informal and direct, not formal, ceremonial, and indirect. Of course, what evangelicals tend to ignore (or dismiss) is the possibility that Catholic Christians do have a fervent, loving, passionate relationship with Jesus precisely in and through the liturgy, as well as in personal prayer, active service, etc. (They also tend to ignore or dismiss the fact that there are evangelicals who have prayed the Sinner’s prayer, wept tears at the altar, professed faith in Jesus, been baptized, and then gone through the motions with church attendance for many years without having a real relationship with Jesus Christ.) They preach on the distinction between “Christianity” and “churchianity” as if “Church” is a dirty word. The fact is that there is no necessary opposition between church and Christ. What do they think they’re doing on a Sunday morning anyway?
Second, “personal relationship” implies a one-on-one relationship with God. This is a crucial difference between Catholicism and Protestantism in my view. Catholic teaching emphasizes that we are part of a people, a Church, a family, a community. We do not preach “me and Jesus” but emphasize that our faith and our actions are those of people in community.
Finally, there is a ceremonial aspect of “personal relationship” that is not taught directly but strictly adhered to. That is, if you were baptized as an infant and grew up in the Church, you still need to profess faith as an adult and be re-baptized. It doesn’t matter how many times you’ve professed your faith on Holy Saturday and been sprinkled with water in renewing your faith, you don’t have that personal relationship with Jesus until, sometime after you are old enough to decide for yourself. You need to make a decision to follow Jesus and invite him into your heart. Better yet, you should be baptized (after, and not before).
That disqualifies any life-long Catholic from having a “personal relationship” with Jesus, because, according to evangelicals, the starting point for that relationship is inviting Jesus into your heart as Lord and Savior, and infants can’t do that. “Believe and be baptized” defines the order in which profession of faith and water baptism are to be done. I must admit that that a deviation from this – baptizing first then personal belief following – are unacceptable.
So, don’t be fooled by the simple slogan. Catholics cannot have a personal relationship with Jesus. I guess Episcopalians and even Lutherans are in the same boat, even though they think of themselves as Protestants.
I know from long, sad, oft-repeated experience, that no matter how fervently you express your love for Christ, once your evangelical friend finds out you are a Catholic, they will believe that you still lack a “personal relationship with Jesus.” They will ask you to pray the Sinner’s Prayer, but even then it won’t stop until you leave the Catholic Church, get re-baptized, and attend an evangelical church. THAT’S what is meant by having a personal relationship with Jesus and being saved. No exaggeration.