Dylan Thomas originally published "And Death Shall Have No Dominion" in New English Weekly in 1933 ("Dylan Thomas Biography," Poetry Foundation). It was his first ever poem to be published. At the time, Dylan Thomas was only eighteen years old and had a job as a reporter at South Wales Daily Post. He had a collection of penny notebooks that he scrawled poetry and verses in as a hobby and pastime; "And Death Shall Have No Dominion" was one of these poems. Although he pulls the title from the New Testament in the Bible, the content of the poem is not overly religious or particularly Christian in nature. Dylan never commented on his inspiration for the poem; there is no 'back story' to the poem, except for Dylan's interest in humanity and its role in the natural world.
Later he included the poem in his second volume of poetry, entitled Twenty-five Poems. Richard Morton noted in An Outline of the Works of Dylan Thomas that the poems of this volume are "concerned with the relationship between the poet and his environment," particularly the natural environment (qtd. in "Dylan Thomas Biography,"Poetry Foundation). Thomas placed "And Death Shall Have No Dominion" at the end of the collection; the optimism of the poem helped the collection end on a more positive, hopeful note.
"Dylan Thomas Biography." Poetry Foundation. web. 17 July 2012.
Just a minor addition to the answer above: Dylan may have gotten the idea for the theme of the poem from a line in the Bible, Romans 6:9--
Knowing that Christ being raised from the dead dieth no more; death hath no more dominion over him.