The speaker of this poem never gets a name, but we do get to know a little bit about him. We're calling him a him because of the masculine adjectives he uses to describe himself in the original Spanish—for example, "caigo en mí mismo" ("I fall in myself" ). The word mismo ends in the masculine "o," so we know this dude's a, well, dude.
He is the one who struggles to remember his lost love ("I search for an instant alive as a bird," ), and who suffers, dying ("collect / my scattered dust and reconcile my ashes," [530-531]) and is finally reborn ("it tore apart my closed lids, / cut loose my being from its wrappers, / and pulled me out of myself to wake me / from this animal sleep and its centuries of stone," [570-573]).
There are clues that the speaker has lived or spent time in Mexico, Spain, France, and the US—he has memories of spending time with girls in several cities in those countries, as well as of the Spanish Civil War. (See lines 248-296.)
We know he's pretty romantic, and also that he has experienced deep loss. Toward the end of the poem, in Lines 469-518, the speaker almost seems to split into two, as he asks and answers questions with phrases set off by dashes.
The thing is, this is when the speaker is getting into some deep meditation and observing himself with his mind's eye, and also when he starts to get into a communion with the whole universe. So it's safe to say it's still the same speaker, just a little more enlightened. The most important things to this speaker seem to be youth, memory, death, and rebirth.