This is obviously a very broad question, but here are a few thoughts that may be helpful. As dmckee points out in a comment, it's difficult to define consciousness. However, consciousness clearly requires computation, and computation is something that physics can address.
There is a psychological arrow of time: we can remember the past but not the future. Physics is sort of able to explain this, in the sense that the problem of explaining the psychological arrow can be reduced to the problem of explaining the thermodynamic arrow of time. Thought and consciousness require computation, and computation has thermodynamic consequences. For example, erasing a bit reduces the entropy of the memory.
There has been some work on whether it is physically possible, given optimal exploitation of the universe's resources, to carry out a computation with an unlimited number of steps. The answer depends on cosmological parameters, and appears to be no for our current cosmological models (Dyson 1979, Krauss 1999). This can be interpreted as a proof that all consciousnesses are mortal.
Typically if we talk about consciousness, we assume that the consciousness is able to make observations. In relativity, this imposes certain constraints on the types of systems that could be conscious. For example, Gorini 1971 gives a no-go theorem for superluminal observers in special relativity, which makes it appear that tachyons, if they existed, could never be the building blocks for a conscious observer. In quantum mechanics, the Copenhagen interpretation ascribes a special role to observers and measurement processes; however almost no competent physicists (an exception being Roger Penrose) believe that this implies anything special about consciousness.
None of this addresses any of the classical religious or philosophical issues; or how to define consciousness as distinct from computation; or whether self-awareness extends to dolphins, mice, or cockroaches; or anything about strong AI. As far as I know, physics has very little interface with any of those matters.
Dyson, Time without end: Physics and biology in an open universe, Reviews of Modern Physics 51 (1979), pp. 447–460, doi:10.1103/RevModPhys.51.447; described at http://math.ucr.edu/home/baez/end.html
V. Gorini, "Linear Kinematical Groups," Commun Math Phys 21 (1971) 150, open access via project euclid: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103857292
Krauss and Starkman, 1999, Life, The Universe, and Nothing: Life and Death in an Ever-Expanding Universe, http://arxiv.org/abs/astro-ph/9902189