More specifically,
If consciousness is a property that can be explained wholly in terms of physical matter, what are the precise physical characteristics or behaviors of things that distinguish conscious things from non-conscious things?
I'm not convinced that consciousness is a property. The way I conceive of it, it's a function, a behavior (or a whole collection of interacting behaviors) performed by a suitable data processing system, the human nervous system in our case.
That's pretty vague, and it's the job of neuroscience to better explain it.
I suspect that ultimately, consciousness reduces to causality. We observe protozoa reacting to changes in pH in their environment and swimming the other way. So we can say that they were "aware" of it and responded appropriately. In the case of a single celled organism, I doubt very much whether there was any "mind" involved or any phenomenal consciousness, just chemical pathways that mechanistically caused the innate response. I perceive protozoa like little machines.
As we ascend the phylogenetic ladder, we see organisms responding to their environment in more and more complex ways. They can recognize predators and food. They can recognize sex partners and (usually, not always) distinguish them from food. So they start performing decision processes on the information provided by their senses.
At some point we see them starting to learn from experience. They don't make the same mistakes over and over.
And at some further point, they seem to gain the ability to think about their own thoughts. Or put another way, they can make decisions about their own behavior and decisions, as opposed to simply reacting to sensory stimulation.
And self-consciousness starts to appear.