"The Love Song of J. Alfred Prufrock" is a modernistic poem that is written in the style of stream-of-consciousness writing. That means that Eliot has Prufrock spill his thoughts onto the paper without editing or filtering them--whatever pops into Prufrock's head, in whatever order, is what gets put down, whether it is in logical sequence or not. The poem follows the stream of thoughts that Prufrock has. Because of this style, because of Prufrock's wandering thoughts and fixation on the main issue he is stressing about--asking a woman a serious question--we can assume that he isn't writing the poem for a woman. If he were, he would organize it more, edit out superfluous mind-wanderings, and probably go about it much differently. He would be more specific about his feelings for her, and clarify the question that he wanted to ask.
My assertion is that he is talking to himself--he is putting down his thoughts as they occur, as a sort of cathartic exercise, relieving some of the strain of his stressful situation. He is, almost, seeming to puzzle out the issue for himself on paper. He starts off talking about how he wants to ask the woman a question, and he goes over the potential risks involved. Then, he wonders whether "it would be worth it, after all" to ask her, when she could totally reject him and he could be misunderstood. He seems rather indecisive, and like he is trying to weigh the matter before moving forward to action. By the end of the poem though, he has talked himself out of talking to her; his thoughts have led him to ponder the risks and conclude that they are greater than the potential benefits.
The poem feels like he is mulling the issue over in his mind, almost like a journal entry where he sits down to clarify his thoughts a little bit. For this reason, it is not written for other readers, but for his own personal benefit. Again, if it was written for others, it would be more clear, focused, organized on-topic and specific. But when we are rambling through our own thoughts and memories, we don't need to be specific--we know what we're talking about. We don't need to be focused or organized--we know how we got to that thought. For all of these reasons, I feel that Prufrock is talking to himself i this poem. I hope that those thoughts help; good luck!