When you're through with something or someone, it means you have given up on them. It has a close meaning to the expression I'm done. For example:
_ I'm through with taking the bus. I'm buying a car tomorrow!
_ I'm done with taking the bus. I'm buying a car tomorrow!
So, in this song, I'd say 'I'm through' would mean I've given up.
And as it approaches
I can't believe I'm through [I can't believe I've given up]
Oh, How I've tried for a life,
Yes a life I thought I knew