Who are the people who wrote these? Over what time span were these written? I know from documentaries etc that the historical existence of David is not confirmed - how come so many are attributed to him? To a layman, many of the Psalms attributed to him do appear to talk about more or less the same sort of thing using similar imagery.
The short answer: This is a horribly difficult problem with no good answer.
Hossfeld and Zenger (I'm always tempted to say und instead of "and" there...) have a great three volume commentary on the Psalter. It's published by Fortress in the Hermeneia series. (We probably should mention Dahood's Anchor Bible commentary here, but note that it's basically just an exercise in Ugaritic in the Psalms.)
First things first: we should likely "throw out" the Psalm titles when we consider things like authorship, because Psalm titles were likely appended at a late stage. Psalms were not likely written by David or Solomon. There's certainly a possibility they were. (I'm convinced that D and S existed, so perhaps they did some writing as well.)
The reason why so many are attributed to him is because he was understood as a musician (note how he is depicted in 1 Samuel 16:23). Oftentimes these Psalm titles also include references to music leaders or to other ambiguous musical terminology.
As far as dating is concerned, we can attempt to use linguistic data to date them (I've always thought that 'wining and dining' is the best approach to dating, myself), but even this can be tricky. Some Psalms exhibit what we call "Aramaisms", or Aramaic features in an otherwise Hebrew text. These could, in many cases, also be construed as linguistic features peculiar to Northern, Israelian [sic] Hebrew, however. Sometimes we can attempt to date certain psalms on the basis of their content. For example, I might date Psalm 74 to the Exilic or Post-Exilic period (i.e., after 586 BCE) on the basis of the fact that it is most likely describing the destruction of the temple by the Babylonians. On the other hand, we have something like Psalm 104, which is strikingly similar to the Egyptian Hymn to the Aten...
Ultimately, we have to take pretty much every Psalm individually, approaching its content (literary and linguistic) with caution. Various Psalms could have been cast with archaizing features, while others could be perfectly reflective of the time in which they were written. Really, it's quite difficult to tell in most cases--but certainly an exciting and interesting topic with which to engage!
Recently I saw an article in Biblical Archaeology Review titled "Can Abecedaries be used to date the psalms?".
According to this article, the order of the Hebrew alphabet has changed, and some psalms (not sure which) are written in alphabetic order (i.e. the first line starts with Aleph, second with Bet, and so on - it's not exactly like that, but you get the general idea, see Psalm 119). These psalms are called acrostic.
An example of an acrostic poem in the Bible is Proverbs 31:10-31. The stanzas follow alphabetic order: They start with Aleph, Bet, Gimmel, Dalet, He, etc. For comparison, see the Hebrew Alphabet.
Of these acrostic psalms, some have a distinct alphabetic order; specifically, letter Pe starts before Ayin, instead of Ayin coming before Pe. This means that such psalms could be dated to a time where the Hebrew alphabet matched those psalms' stanzas order. Some archaeological digs like Kuntillet Arjud, have uncovered old abecedaries with said order.
Here's an abstract for the relevant scholar article.
http://m.jot.sagepub.com/content/38/4/471.abstract
Various abecedaries with the unusual order of the letter pe preceding the letter ayin have been discovered in ancient Israel. These abecedaries date from 1200–600 BCE. The pe–ayin order was already known from the acrostics of Lamentations 2–4, and from the Septuagint text of the acrostic of Proverbs 31. Scholars have not appreciated the significance of the pe–ayin order as an aid to dating the Psalms. The acrostic of Psalm 34 makes much more sense under the assumption that the pe verse preceded the ayin verse, and it seems that all the acrostics in the first book of Psalms (9–10; 25; 34; and 37) originally reflected the pe–ayin order. The acrostics in the fifth book of Psalms (111; 112; 119; and 145) reflect the ayin–pe order, an order that seems only to have arisen in Israel in the post-exilic period. The differing orders in the first and fifth books help to confirm an early date for the first book and a late date for the fifth book.
TL;DR: dates and attributions of Biblical authorship happened for many reasons, but most of those reasons have absolutely nothing to do with who actually wrote them and when. It's important to know that both date and authorship carried political and religious consequences, and therefore the practice of obscuring and mis-attributing authorship, and backdating texts, was so incredibly common that it even has a jargon term all its own. Your question therefore brings up interesting points about religion's influence on history, and it brings historiography itself to the forefront; when Late Antiquity or Medieval historians and scholars went to work, their goals were a bit different than when modern historians do their thing. NEVER trust an attribution based solely on an ancient or Medieval tradition. As /u/Husky54 demonstrates below, convincing arguments will be empirical, and based on linguistic or cultural analysis, not on traditional sources.
Long version:
This is not an answer, just a bit of side-commentary; when it comes to attributions of authorship and dating Old Testament Biblical material, it's important to know that there's a religious tradition that asserts that the scripture is fixed and unchanged past the point of the destruction of the First Temple in 587 BC.
Wikipedia notes: "There is no scholarly consensus as to when the Hebrew Bible canon was fixed: some scholars argue that it was fixed by the Hasmonean dynasty,[4] while others argue it was not fixed until the second century CE or even later.[5]"
But the Rabbinical and religious belief is that there can be no "new" scripture. Once the Temple fell, God was done talking directly to Man. The end; no arguments! There are no new revelations. That's a potential 800-year difference between when canonical texts were written versus when they were SAID to have been written. That's a huge gap! (And we'll get to the non-canon, esoteric, and mystic texts below.. To say nothing of the Apocrypha)
So there has been a strong motivation throughout the centuries to deliberatly mis-attribute later writings to earlier authors, so that they still appear religiously legitimate (i.e. to avoid an accusation of heresy, which can have extreme consequences). This is a question of scholarly priorities. It's only in modern times that factual accuracy has been a primary goal in historiography in general. Through the several thousand years between the Old Testament's first mutterings and the present day, scholars have wanted foremost to shape history, rather than uncover it. History, geography, and even travel writings, were genres which functioned more like Science Fiction does for us now: to prove a moral or political point, or to provide a fable to change people's thinking. It was discursive, not descriptive.
This book does a great treatment of the Greek Travelogue genre: The Edges of the Earth in Ancient Thought -- in this text, it's easy to see how history, sociology, geography, travel, and what we would now call anthropology all fit into this polemical paradigm, in the hands of Ancient World writers.
A lot of historiography, and Biblical scholarship as well, was done with the express knowledge that whatever we write now, later generations will have no choice but to take as fact. This appears to be a cross-cultural phenomenon, and we even do it today. Re-writing who did or said what, and when, helps people make propagandic arguments that will serve them in the future. Biblical scholarship is prime power-grab territory, and it's more fantasy than history, right up until the modern era, when scholars have begun to value accuracy foremost.
You can also imagine that attributing contemporary writings to ancient and famous authors is a way to short-cut your own writings into religious, literary, or cultural respectability.
The technical term is pseudepiagrapha and it is extremely common in Late Antiquity and the Medieval period.
This is also a recurring theme in Jewish mystic and esoteric religious traditions; books written in the Early Modern period in Europe would be professed as "ancient texts" and given famous authorship. It would have been a major heresy to claim new scriptural revelation, so primary texts of the Kabbalah are either attributed to ancient sources and considered pre-Destruction as a matter of convenience/not getting thrown out of your community as a heretic, or they're presented as "commentary" on existing Tanakh verses. "In the medieval era Jewish mysticism developed under the influence of the word-number esoteric text Sefer Yetzirah. Jewish sources attribute the book to the patriarch Abraham, though the text itself offers no claim as to authorship."