Kill enough enemies using Avenger Strike to cross the points threshold and this challenge will be completed. Any difficulty, enemy, or map will count towards this challenge.
Don't waste skill points from other skills trying to make Avenger Strike stronger with power increases - this skill is only affected by melee damage increases. This includes boosters, equipment, mods, skills, or anything else that increases melee damage. Avenger Strike itself can increase its effectiveness with increased skill points but outside of this only melee damage increases can make Avenger Strike stronger. Be aware that anything that reduces melee damage will also reduce the damage of Avenger Strike.