This is a very weighty question. I will start this by stating that I am a medievalist and therefore my methods are based on the problems and tactics of a pre-modernist and may differ somewhat from other historians who work in modern disciplines. That being said, I'll try to give a fairly broad over-view.
There are a variety of ways to determine "reliability" but before I even talk about that it is worth diverging a bit.
It is always important to remember that historians are not just using sources to create a "narrative", they are also analyzing sources as a window into the mindset and world of the source and the author. In this case the question of reliability is often less important. This is especially true with pre-modern work where older notions of the "usefulness" of a particular source (take poetry as a good example) have been thoroughly revised once we remove the insistence that a source is only good as a dispensary for facts. I addressed this somewhat in another post where someone asked about how we deal with the "fantastical" here.
Now, lets talk about how we figure out how "accurate" a source is. As I said, there are a variety of ways, some easier than others. So, in no particular order...
We can compare a source to other contemporary sources: The more sources we have, the better.^1 We look at a variety of sources and see how often they converge or diverge on a topic. If ten documents say that x happened, and one document says that y happened, typically we will err on the side of x (though not always). On the other hand, sometimes we can use different documents to patch holes in other documents. If document A says that an event occurred but gives no location, and document B gives a location but less info on the event in question reading both allows for a better picture of the event. We also look at how documents "depend" on each other. If document A is the source of documents D, E, and F but not of B or C, than D, E, and F may be less useful for "corroboration" of A. For modernists this also can involve talking to people, an ability that pre-modernists might envy but which is just as (if not more) complicated as reading a variety of sources. It has been shown time and again that the human mind is not a "recorder" so much as an "interpreter".
We date documents through a variety of methods: Sometimes this is simple; a document may explicitly say when it was written. Other times we have to rely on a variety of techniques, from "common sense" (if a document doesn't mention anything that occurred after x date, it was probably written before it) to technical skills (the handwriting or techniques used to produce the document provide info). Generally (though not always) the closer to an event a source is the better.^2 By dating a source we can judge how reliable it is based on whether the person could have been an eye-witness, or talked to an eye-witness, or whether they are receiving stories passed down through a generation or something they read about etc.
We look for other data: Archaeology, art history, geography and other forms of data all have the ability to corroborate or expose weaknesses in a source. If Document A says that a town was burned to the ground and archaeologists find no evidence of a burn layer, we have to question the document, at least somewhat. We can also use skills like paleography, linguistics etc. to look for things like forgery and anachronisms that might make us question the accuracy of a document.
We examine the document closely via critical reading: We read for bias, we read for the author's goal, we read for the author's position or station in life, we read for context clues, so on and so forth. By analyzing the document as a written piece by a thinking person we can often come to ideas about the "reliability" of a document. Perhaps the author has a bone to pick with King Whosit, and therefore he chooses only to talk about the negative things that occurred during said king's reign. Perhaps a person is trying to hide atrocities committed by their side during a war, or to play up the suffering their own side suffered. Perhaps the author uses a lot of biblical language, or classical quotes. This may lead us to question the reliability of an account. Did the besiegers actually have "catapults" as we conceive of them or did the author use that word because a famous classical work describes them at a great Roman siege? In many ways critical reading is the most basic and the most important skill a historian uses. It is often the most difficult as it requires both a deep understanding of the culture and people in question and the ability to pick at arguments and read deeply and questioningly.
We use "common sense": Humans can't fly. We know this. It is a fact. If a document says that humans flew, at the very least we question that particular part of the document. This doesn't mean we discard the document whole hog, but we do have to deal with the fact that humans can't fly. Modern knowledge of science can allow us to judge the validity of a document. This is especially true in the pre-modern world where miracles and myth are not particularly distinguished from history.
So there you go. A by no means exhaustive list of some techniques and tactics for determining a document's reliability. Others may have more to add, or may poke at me a bit. Happy (and critical) Reading!
Footnotes!
1: A side note, this is one place where modernists and pre-modernists face different challenges. Pre-modernists are often forced to make due with a very limited number of course, milking each one for what it will offer. Modernists, on the other hand, are often faced with an over-abundance of sources and have to figure out ways to sift through the mass, to narrow their focus or to eliminate non-useful data.
2: Another side note. There are cases where a later document will be more reliable than an earlier one due to "clean" transmission. In other words, although the source is written down later it is actually a copy of an earlier work that has been lost.