These are part of the lyrics of the fifth ending theme of Hunter x Hunter (the 2011 animation).
It seems 「生かんとする」 means 「生きようとする」:
生く(archaic form of 生きる) → 生か (未然形) + む(ん)とする
but I'm not sure whether it is a relative clause here.
Does the second line mean "my nature of trying to live", like, as part of my true nature, I try to live? Or maybe 「生かんとする」 isn't a relative clause at all? Since it's a song, I'm not sure.
I don't understand what "to try to exist/live" means exactly, either. Especially since
volitional + とする is supposed to carry the nuance of being unsuccessful.
It seems 生く can also mean 生かす (to keep alive), so could 「我が性」 actually be the direct object of 「生かんとする」? Can I use
volitional + とする as a relative clause modifying the direct object of the verb in volitional form if I'm the agent? Like:
the computer I tried to use (and failed)
Related question about んとする: What is the difference between ～んとする and ～とする?