"I got the eye of a tiger, a fighter."
Well, a tiger is considered like a really fierce character. So, I think this resembles courage and initiative.
"Dancing through the fire."
The fire is like her struggles. So she technically "dances" (overcomes) her struggles.
"Cause I am a champion, and you're gonna hear me roar."
The "you're gonna hear me roar" part makes me think about "shout in victory", become better or "break the limits".
"Louder, louder than a lion."
This, to me, is just an exaggeration of the previous line.
Also, off topic, but your English is really good! You wrote this just like a native! Keep up the good work!