That kind of a relationship, a familial one, is the relationship we look forward to as Christians. When we, as believers, will see Him as one can see any other person walking down the street.
There are those who express they enjoy that kind of relationship with Christ now, even some who insist that type of relationship MUST be the norm elsewise one has not yet truly believed. I do not fall into that camp. But I will try to relate what I can from what I understand of Christ thus far.
Jesus said quite plainly, "Where I am going you can not follow, but if I go I will leave a comforter with you" (BPV- BiggAndyy's Paraphrased Version). The person we do get to experience in our lives now is the Holy Spirit, not Jesus (God come in the flesh).
Now I realize all I've really done is move the goalposts a little bit and not answered the question, merely rephrased it; "How does one experience the Holy Spirit?" I will give you five ways the Holy Spirit either is present in our lives, or is accessible to our lives (they are short and certainly not exhaustive in their quantity or quality).
First is through Scripture (the Bible). If one expects to have any kind of meaningful relationship with anyone it must be built on a foundation of both knowledge and trust. The Bible is, of course, our first and primary source of knowledge of all things spiritual. But it must also be a trustworthy and faithful account of both history and teaching. If we fail to grant these first two provisos then everything else just can't make sense.
Second is through prayer and meditation on the words we've just read. After all, if one must study in order to learn something like chemistry or sociology, merely a few cursory glances at several pages or chapters of the Bible will impart no information at all, much less understanding. The Bible is where we are introduced to Christ (and the Holy Spirit), prayer is where we can make our introduction to Him.
Third up is participation in corporate worship. I know that a new phenomena has popped up in the American Church, the Seeker Sensitive Church. But Sunday Worship should be a time where believers come together to bring worship to Christ. The worship service is not a place to try to "convert" someone to the faith. It's like taking an eighth grade biology student to an autopsy; it's just not the time or place for such a person to be. But corporate worship is valuable to the Christian for edification (building up the faith of some, restoring the faith in others) and is a place where we get to express, in public, our love and dependence on the Holy Spirit for our times of trouble and even for the mundane.
Fourth is small Bible study. Where those less well versed in scripture can go to learn, ask questions, and be reinforced in their walk with the Holy Spirit. Mush less formal than true worship, it's where we get to "practice" our Christianity with others.
Lastly, we experience the Holy Spirit (and Christ) through Christian relationships. Having friends who are fellow believers, to pray with and hold us accountable when we fail, and to be an additional resource for God to have into our lives. No Man is an island, and no Christian is a Christian unto themselves. Without others, like the coal taken from the fire, we grow cold, distant, and can feel like God has abandoned us.
Christianity is a faith we give away, because two things collide at the cross, our sin and God's mercy. No where else can those two be dealt with except at the foot of the Cross, where forgiveness ultimately flows.