BATTLE AXE

I unlocked this in easy mode. I saved all the civilians and didn't take any damage. I don't know if time is a factor but I didn't rush through it, but I also didn't go back to pick up any coins dropped by enemies behind me.

Edit: It seems like the main factor is not getting hit. Playing in co-op makes this easier. Each player is awarded their own grade, so you could get an S and the other player an E and it'll unlock for you.