They're as reliable as any other historical sources, but you have to read them on their own terms and understand that they do have their own projects that they're trying to accomplish.

All historical sources were written to do something, and this usually involved more than just presenting 'the facts.' So you have to realize what they are and read them with that in mind: sermons that mention the vikings may use them as a moral lesson intended to prompt the hearers of the sermon to change their lifestyle, while the Anglo-Saxon Chronicle wants to make Wessex look like the best kingdom ever, and presents the viking invasions in a way that makes Wessex's eventual victory look really fantastic.

But the question of whether English sources about the vikings are biased isn't the most helpful. They are certainly biased, but everything - including this reply - is biased, and we couldn't do history if we limited ourselves to unbiased sources because everything written and read by a human is influenced by that human's unique perspective. The important question to ask is: what is the author trying to make me believe, why and how is s/he trying to make me believe it, and what can I learn about the time when the author was living and the events the author is describing by seeing what s/he is trying to make me believe? You don't test to see if the source is reliable or not; you test to see what the author is doing, and use that knowledge to make judgments and interpretations about the conclusions you can and can't reasonably take away from their text.

In the case of stories about viking aggression, we can learn a lot about English political formation (because many of the texts that describe the viking invasions are really stories of how England formed as a united political whole), English beliefs about the vikings (though we have to realize that the vikings are often set up as an 'other' through stereotypes and distortions that might tell us more about English beliefs than what the vikings were actually like), and even a bit about who the vikings might have been on their own terms. And we can test our interpretations of the English evidence against other kinds of evidence (texts from outside England, archaeology, etc) to see how the picture they present compares with other possible versions of the past; taken together, these multiple perspectives give us a more nuanced, complex, and multi-faceted view of who the vikings were than if we just looked for the most 'accurate' sources and trusted them.

But you're right that we don't do this by taking the sources at face value as presentations of 'facts' - we have to read them very carefully to see what they're really about before interpreting their contents carefully. It's just a lot more complicated than looking for their 'bias'; we have to reconstruct much more of their contexts and see how individual writings fit within the worlds in which their authors were living.