Should I kill Knight Lautrec of Carim before or after he kills Anastacia of Astora

After. You can revive her anyway, and it's much more fun to do his invasion fight. Also, you have to invade him to get his armor. Just make sure not to reinforce estus after you get Anastacia's soul, not until you revive her at Firelink. Else you might end up spending it and being unable to use the Firelink bonfire for the rest of the playthrough.

ok cool thx

Better to play through the story line the first time. but in other playthrough, you can just push him off the edge at firelink, then reload for his ring. Not particularly advisable unless you need the ring as soon as possible of course.