"When Death Comes" is a free verse poem. That is, it doesn't follow any particular formal structure of rhyme or meter. Still, there is a definite rhetorical structure: 'When _____, I want to ____. Therefore I ____.' Then it repeats the form partway: 'When ___, I want to ___. When ____, I want to ____.' It looks funny written like that, but that's essentially the movement of the poem. Of course, our speaker draws out most of those parts, linking a bunch of clauses together, so that the basic structure is a lot less obvious, and the rhythm is a lot more interesting.
That rhythm draws heavily on repetition and piling on successive clauses. Almost every line is either beginning with a phrase that was used earlier, or establishing a phrase to be repeated later. Each of the following is repeated at least once in the poem, as the beginning of a clause – "When death comes," "I look upon," "and I," "and each," "when it's over," "I was," "I don't want to." Aside from sounding good, the effect of this repetition is to make the language feel very rich. It's like doing variations on a melody. If the exact same phrase were being repeated it would probably get boring, but by starting at the same place then reaching in a different direction, it helps to link ideas together while also expanding their reach.