In order to unlock this achievement you must rescue Zote in Greenpath the first time you progress through the area at this location:
If you leave him to die, you can return to this point later and earn the Neglect achievement. You cannot get the Neglect and Rivalry achievements on the same save file.
So rescue Zote from the Vengefly King and he will relocate to Dirtmouth. His next location is in City of Tears:
You can find him next in Deepnest:
Free him here. What an ungrateful chap.
With this all done, proceed to toe Colosseum of Fools in Kingdom's Edge:
Partake in the Trial of the Warrior. If you properly progressed Zote, then he will be a bonus final boss after the two Gruz Mothers. He is absolutely inept and puts up zero resistance. Attack him until he is defeated and this achievement will unlock.
I'm not 100% certain but I believe you must have rescued Zote from Greenpath and Deepnest BEFORE completing the Trial of the Warrior. It's possible that you could replay the trial after rescuing him in Deepnest and he will appear but I'm not sure so to be safe, rescue him before doing the trial.