The direct translation would be : "Why do you fall in love alone and then break up alone"

'I'm hurting alone' is not correct. You are right. 이별하다 means 'to part/to seperate/to break up' By the way, "Why am I loving alone <while> you left alone". is slightly incorrect. The sentence dosen't mean <while> but just <and>. > Why I love alone, why I have to say good-bye alone(/why I seperate alone?) We can't know the exact intention and reason cause it's a kind of metaphorical(or emotional) expression though, we can guess that he is in a onsided love or she just went out(played) with him without taking him seriously.