On the most basic level we're interested in how such documents survive, which means tracing the manuscript tradition and trying to find what we call the "archetype." That is the earliest form of the text that can be reconstructed from extant manuscripts. This is the same as every ancient document (to be honest, the short answer to your question is "just like everything else"). This is NOT - NOT! - the original. We call that the "autograph" and at least for my field none exist anymore.
After we have a secure manuscript tradition, we get into issues of emendation and interpolation (fixing gibberish passages and finding things that don't belong), and people are still arguing about this stuff. Often theology comes into play (I reject such and such a passage because I have a doctrine that doesn't agree). Barth Ehrman has a great book on this process call Misquoting Jesus. He's not an uncontroversial scholar, but his treatment of the textual criticism of the Bible is the easiest to access I know of. A more complicated book that does a better job of teaching actually text crit is by Bob Renehan. I have it around here somewhere but I can't find it at the moment. I think it's title is Greek Textual Criticism. You'll need to know a fair bit of Greek to make any sense out of it.
I think the biggest thing to remember is both the Bible and the Koran developed into their current form in ALREADY Christian and Islamic societies - they're reflections of their time that Christians and Muslims wrote, rather than documents that produced new movements. Personally I think questions of authorship are not so important once one recognizes this. It's cool to know Paul wrote this letter and not that one, but the Christians that compiled what we call the Bible today didn't have the same concern for authorship as we might today.
Then we go about checking other texts from the same period - for instance we know from the Res Gestae Augustus did not make a census of the whole world (Sorry, Luke!). Apart from being a completely ridiculous proposition for a pre-industrial society, Augustus would certainly have mentioned it, and he doesn't. I should add for the Bible (I am less familiar with the Koran), we also need literary scholars; there are many different genres in the Bible that need to be interpreted in different ways. There's poetry, plays, history, mythology, etc etc. Identifying the correct genre is a key element if putting these documents to use.
I'm actually going to disagree with /u/legalaction on his understanding of the Quran, particularly when he says that "...the Koran developed into [its] current form in ALREADY...Islamic societ[y]."
Because the Quran is such a markedly different kind of text than the bible, that's not entirely true. The bible, whether you accept an earlier date or a later date, was not a concious project of Jesus. It was, as /u/legalaction says, a text produced by Christians, living in a Christian society (even if they were a minority), and writing about past events.
That's not the case with the Quran. While there are issues about compilation, and how closely this resembles early fragments of the text that survive, it's nonetheless reflective of an actual project that Muhammad himself undertook and that many scholars believe may have actually, particularly towards the end of his life, may have started to write down. I personally think that the Uthmanic Codex, despite issues over the order and very obvious issues about precise features, is probably very close to what Muhammad would have recited to his followers.
All of that said, there's a separate issue in terms of how different the Quran is from the Bible, which is that if you've ever read the Quran it's not actually very historical. A good really short example is Sura 105, "The Elephant"
1: Hast thou not seen how thy Lord dealt with the owners of the Elephant?
2: Did He not bring their stratagem to naught,
3: And send against them swarms of flying creatures,
4: Which pelted them with stones of baked clay,
5: And made them like green crops devoured (by cattle)?
Now, the traditional sources relating to this and the early Islamic historians say that this verse refers to an event where an Ethiopian king marched on Mecca with war elephants. But the fact is that as historians we have no other verification that any such event ever occurred. And that's for one of the most straightforward historical incidences in the Quran.
The rest of the Quran simply doesn't deal with contemporary or past events the way the bible is. The Romans (Byzantines) are mentioned in passing I believe two times in the entire text. The stories of the prophets are often condensed. The Quran itself is closer in length to the new testament. It's not arranged by chronology, or theme, or subject, but by the length of the Sura.
As such it's a crucial historical document, around which, following /u/legalaction there is extraordinary textual analysis, and scholars of Islam have tried to glean as much historical information from it as they can, but unlike the bible it doesn't really make all that many historical claims.
For that you really need to dive into Tafsir and the traditional Islamic histories, the historiography of which tracks much more closely with biblical historiography.
Edit: if anyone wants to read more, Neal Robinson's Discovering the Quran is the best introduction I've come across, covering multiple views from traditional religious, early western academic "standard," the revisionists, and the counter-revisionists.