His lifelong / continued / continuing obsession was taking revenge on me. There's no need to use 'hurt' here because the word 'revenge' means 'hurt or punish somebody'...

That's a lot of "ing". One revenges oneself against, on or upon someone else. By saying "his pursuing obsession was hurting and revenging me", you've backflipped at the last moment. "Revenging me" means you don't receive the hurt, but someone else is hurt in return, to your advantage. How about: "his houndlike obsession was vengefully hurting me." I've cobbled 'hound-like" together to keep the image of pursuit, eg. hounds after a fox, or "to hound" someone.

Hi it would be better to write: " His obsession was pursuing, hurting and seeking revenge against me."