Think of it this way. Itachi killed everyone he loved and he was not strong enough to protect them. As a matter of fact, he wasn't even deemed strong enough to be killed along with them. Now, years later, he has other poeple he cares about. What if Itachi (or someone else) came back and decided to kill them too. What if he wasn't strong enough to protect the ones he loved a second time. So in his mind he has to become stronger at any cost because if he doesn't he will lose everything again anyway. Better to lose and have a chance to regain than just to lose right?
The way he sees it, If he doesn't kill Itachi he will lose those friendships anyway. So if he kills Itachi, at least his friends would be safe and he would have avenged his clan. If he doesn't then he is as worthless as Itachi says he is.