What is the Biblical basis for the use of extra-Biblical books in the study of Christ?

The Wesleyan/Methodist position on this would be that it is possible for two equally learned persons to view a scripture and come to entirely opposite conclusions. For example, this can be seen in the doctrines of Predestination. Both an Armenian and a Calvinist will claim a "plain and obvious" reading of scripture, yet will arrive at opposing conclusions regarding the nature of God and the free will (or lack thereof) of mankind. There are many other doctrines and passages like this.

Therefore, only using scripture (sola scriptura) will not allow us to see or be elucidated by the truth of God as something more is required. This is even somewhat Biblically supported as the Apostles were ordered to preach the Gospel, not go into all the world and read scripture to all nations at the Great Commission.

The Wesleyan tradition holds that because mankind is sinful and fallen, they cannot help but filter it scripture through that fallen corrupted lens in seeking truth and understanding scripture when attempting to draw nearer to God. Instead of pretending like this isn't the case, by recognizing and attempting to minimize the corruption inherent in our ability to understand what God is trying to say to us, we can better draw closer to God.

This is not to say that Scripture should not be the foundation for this. The Wesleyan tradition holds that all doctrines should start from a basis in scripture first (prima scriptura). Then, when disputes and questions arise regarding a scripture, reason, tradition and personal revelation should be weighed equally to aid in determining the truth that God has for the reader. The reader should consider the view or doctrine philosophically and logically to ensure that it is sane and consistent. The reader should also review traditions and historical teachings of the church. Thousands of years of very bright theologians may have considered a given issue and prayed diligently about an issue. Therefore their works should be considered and weighed. Finally, we server a God who is risen and alive and still speaks and works in us. Therefore, we are capable of receiving revelation from God. Because we are fallen however, this revelation may not be of God, which is why the other 3 squares of the Wesleyan Quadrilateral can override any one square (except the scripture square). Likewise, the church is not incorruptible. Were tradition without flaw, we would not have Protestantism. Furthermore, it was traditionally held for a very long time that slavery was Biblical. Just as with the other two squares our logic is obviously flawed and corruptible.

Extra-biblical writings allow us to record tradition and to engage in debate in order to pursue reason in regards to interpreting scripture.

These things are all something we will do when interpreting scripture regardless of whether we realize it and recognize it or not. We all read scripture in terms of our tradition and the church we were raised in and the beliefs we were taught. We all believe that which we believe God is instructing us to believe and we all believe in that which is logical and sensible. Embracing this allows us to have a healthier faith.

Finally, the Bible was written in a language that is different than ours at a time which was different than ours. Unless we are all to learn fluent Biblical Hebrew, Greek and Aramaic we need tools - extra-biblical books like lexicons - in order to understand scripture. It uses idioms which have been lost to time and has pop-culture references which are lost on a modern westerner. We therefore need commentaries to put scripture into context to help us to understand it. Yet these are all extra-biblical sources.

The best comparison that I can make is to Shakespeare. I have a very difficult time understanding the Bard without a good dictionary and a few good commentaries. If I do not understand what a Montague or Capulet is, I simply do not understand why the love between Romeo and Juliet is a problem and forbidden. I need a commentary to help me understand this. And this is when I can even understand Romeo and Juliet are saying at all - and Shakespeare is in my native language and only 400 years old! Now imagine that the Bible is in the old-English equivalent of Hebrew and Greek and is 1600 to 4600 years older than Shakespeare. Clearly it's not just as simple as a "plain and obvious" reading of scripture.

The question as stated is a little broad, but I will try to point to some Biblical bases for using extra-Biblical sources to aid the common Christian endeavor to follow Christ.

I am answering from the perspective of my Catholic faith, so perhaps I would not whole-heartedly assent to an unstated assumption in the question as stated, that all beliefs need a Biblical basis in order to be valid. A Catholic would also assent to the two other sources of Tradition, loosely defined as the oral teaching of the Apostles as expressed in the unanimous consent of the Church Fathers, and Magisterium, which serves as arbiter between conflicting interpretations of Scripture and Tradition. Given this brief caveat, I will try to confine myself to Scripture to show the reasonableness of using extra-Biblical sources.

Consider the very last verse of the four Gospels, Jn 21:25:

But there are also many other things which Jesus did; were every one of them to be written, I suppose that the world itself could not contain the books that would be written.

We have a Biblical example of this in Acts, when Paul says to the Ephesian elders (Acts 20:35)

In all things I have shown you that by so toiling one must help the weak, remembering the words of the Lord Jesus, how he said, ‘It is more blessed to give than to receive.’”

Where did Paul hear these words? Evidently, from one of the Apostles. Granted that Acts is now part of the Canon and not "extra-Biblical"--though I would caution that it is naive to ignore the human process by which the sayings and deeds contained in Scripture were gathered and redacted--this example still sheds light on the broader question: why read anything outside of Scripture?

St. Paul recommends the reading of Scripture (Col 4:16):

And when this letter has been read among you, have it also read in the church of the Laodiceans; and see that you also read the letter from Laodicea.

...but it is curious that his most often repeated words of counsel are similar to 1 Cor 11:1:

Be you followers of me, even as I also am of Christ.

and (Heb 10:24):

And let us consider how to stir up one another to love and good works.

These last two recommendations involve imitating a person or speaking with words--words which in no way are "Biblical". If Paul encourages me to exhort my brother in the Faith, is it no surprise that ever since then Christians have used the mouth and the pen to exhort their brothers? The epistles of Ignatius, Clement, and Barnabas are some of the earliest examples. Clement, for example, is firmly conscious of this office of "admonishing" when he writes in c. VII:

These things, beloved, we write unto you, not merely to admonish you of your duty, but also to remind ourselves.

"Admonishing" is an extremely "Biblical" action, though it involves our own words. Though we have many records of the words that Paul or Peter used to preach to the crowds, how many times are we simply told that they "preached" and "converted many"? Those words were certainly extra-Biblical.

In the end, the point is ridiculously simple. Christian books are the attempts (some great, some mediocre, some bad) of Christians to fulfill this Pauline precept of "exhorting" their brothers and Christ's command to "preach as you go, saying, ‘The kingdom of heaven is at hand.’" (Matt 10:7) There is no formal difference between preaching by word and preaching by pen. If I can listen to a homily, why can't I read a book? I personally have enjoyed many of the radio broadcasts and books of the Archbishop Fulton Sheen, though I never listened to him in person.

Should we use any source other than the 'one source'?" Of course. Just as I look up on a map to see where Cana is, or read scholarly sources to see what Sadducees believed, or read the Bible in translation (=interpretation), so I will go to people I have good reason to trust--both living and dead--to seek guidance in following Christ, whether it be by word, letter, radio, or book. The "one source" never suggests that is is the "only source."