The achievements for Underdog and Bully are to kill a demigod 10 levels above you, and kill a demigod 10 levels below you, respectively.

Now, the Bully achievement is probally fine, but still a very difficult one to get. Creaming an opponent and denying them so much xp is highly unlikely, but possible I suppose. However, I've never even seen anyone get 10 levels above an enemy demigod.

The Underdog achievement, however, is downright impossible unless it is intentional for the player getting killed. 3 demigods that are 5 levels lower than the one they are attacking can have a rough time killing that demi as it is.

It would be much more reasonable, yet still difficult, if you had to kill a demigod 5 or 6 levels higher than you.