This is pretty simple to do with Lazrael or any character with fighter levels as she gets action surge.
Once you get to 5th level you will unlock Extra Attack which gives you 2 attacks per action. For me this actually unlocked when I did 4 attacks in one round which was simply attack, attack, action surge, attack, attack. But if you need 5 you can also give her two weapons and that should push it to 5 attacks at level 5.
I am not sure if other bonus attack actions like pommel strike would also work for this achievment.