seikun

It's very hard to deal with relative clauses like this in a natural way in something like lyrics because as I'm sure you know English has a much lower tolerance for them and "which" "that" "who" type words. 
Let's first assume the missing predicate here is です and thus the meaning is 一人きり泣いてた私へのシンフォニアです。 and not anything more complex.

There's a couple of strategies you can consider in this case:
1) Changing the form. Adjectives would work much better in English.
"A symphonia for my lonesome weeping self" or some such

2) Flouting conventional sentence structure. For something like lyrics you can get away with dodging these kinds of dilemmas with punctuation...
"A symphonia for me, weeping all alone" or some such
This adds ambiguity but that's not necessarily a bad thing in lyric translations imo

But, since the predicate is missing, you could assume there's a more complex predicate based on your interpretation of the song. Here's one I  just pulled out of my ass without reading the rest of the lyrics.
一人きり泣いてた私へのシンフォニアが流れてきた。
"I was weeping by my lonesome when a symphonia for me started to play (alt: symphonia started to play for me)" or some such
Yes, the translation "breaks" the original sentence structure which tends to scare some translators, but if it doesn't destroy the elements with meaning I don't really think it's a problem; 1:1 translation is near impossible with poetic Japanese. Leaning towards liberal translation in lyrics/poems can often produce a better final product in the target language.

Hopefully this at least helps you consider your options.

"Sinfornia for me who was crying all alone !"
Somehow you think as " a music for a crying person " , and the person is " me ".

Thanks for the replies.