You should absolutely read the book No Country for Old Men, written by Cormac McCarthy. It's a very easy read and short, literary but genre lit, so accessible to us all.
My understanding, after pondering both the book and film, is that Chigurh is really of a "force of nature", brutal, uncompromising and monstrously indifferent. Not just a psychopath, but a "psychopath's psychopath". He kills because that is what he does. However, there may be deeper meaning to the pigeon in that birds are often a metaphor for the soul. Thus, it may indicate that for all his dominion of the physical world, he cannot destroy the soul.
One of the themes of the book, also touched upon in the movie, is the eternality of violence. The book is set against the ultra-violent, modern drug wars, with the feeling that it's never been so bad. (The old Sheriff realizes he's getting too old to deal with it, hence the title.) But then there are passages recounting his grandfather's time, and a bloody vendetta, no less brutal than modern times.
The brutality of both man and nature is a recurring theme in McCarthy's work. Blood Meridian or the Evening Redness in the West is set in that earlier period, the 1850's, in the same general region. The Road, also recently adapted into a film, involves similar themes.
The title of this work comes from the famous Yeat's poem, Sailing to Byzantium, and Yeats definitely has some thoughts on the soul. The poem reads:
That is no country for old men. The young in one another's arms, birds in the trees, those dying generations at their song, the salmon-falls, the mackerel-crowded seas, fish, flesh, or fowl, commend all summer long—whatever is begotten, born, and dies. Caught in that sensual music all neglect, monuments of unageing intellect.
An aged man is but a paltry thing, a tattered coat upon a stick, unless
Soul clap its hands and sing, and louder sing for every tatter in its mortal dress, nor is there singing school but studying monuments of its own magnificence; and therefore I have sailed the seas and come to the holy city of Byzantium.
O sages standing in God's holy fire as in the gold mosaic of a wall, come from the holy fire, perne in a gyre, and be the singing-masters of my soul. Consume my heart away; sick with desire and fastened to a dying animal—it knows not what it is; and gather me into the artifice of eternity.
Once out of nature I shall never take my bodily form from any natural thing, but such a form as Grecian goldsmiths make of hammered gold and gold enamelling to keep a drowsy Emperor awake; or set upon a golden bough to sing to lords and ladies of Byzantium of what is past, or passing, or to come.