You should hit 10,000 regular enemies pretty quickly.

Currently I have one requirement on an Ancient Relic Multi Melta that says I have to kill 500 enemies when below 50% Health. I am a Tank Crusader. I have 13 of those kills. I've had it working for weeks now.

In all reality though, I haven't found a single Ancient Relic Weapon to be worth using over an Archeotech Relic. I've turned almost every Ancient Relic I've looted into Sparks of Glory - Sockets are definitely worth more than Ancients to me.