By itself it's a rather cryptic sentence. Of course that's not surprising in poetry and song lyrics. You have to look at the larger context of the song.
In this case, just go back one and forward one line: "Wake me up inside" and "Save me from the nothing I've become". So the singer is saying, "Wake me up, bid my blood to run." As Kiamlaluno says, "bid" here means to command or request. So the singer is saying, wake me up, and command my blood to run through my body. i.e. force me to be alive and conscious, and not to be a lifeless non-entity.
"Before I come undone": Again, as Kiamlaluno says, "undone" means "defeated" or "ruined". So the singer is begging someone to save her before she is defeated.