When I read the thread title, my immediate answer was "no", but in the OP was the question "did he do the best he could?", and I think he did.
Worf knew from the start he would be bad at this. He sent Alexander to live with the Rozenkos because they were good at being parents. But that had side effects.
I think there was a lot of tragic miscommunication there: Alexander thought Worf sent him away because Worf didn't like him, and knowing this made Worf upset, and seeing Worf upset made Alexander think it was because he wasn't Klingon enough, which made Worf more upset. Alexander couldn't see that what was making Worf upset was Worf's feeling that everything was his fault and a better father would have a solution.
Worf didn't really have a role model for a Klingon father: he was orphaned around the time he was Alexander's age, so he didn't have a Klingon father to base his behavior towards a child that age on.
The Rozhenkos gave him a model, but that was for raising a child who wants to immerse himself in the study of Klingon society, so he had no idea how to raise a kid who didn't want that, and less idea how to make that interesting to someone.
So most of Worf's ideas on how to be a good parent involved stepping out of the picture. He had seen that with others Alexander didn't have the baggage of feeling he wasn't living up to their expectations, and he couldn't figure out how to communicate to Alexander the pride and love he felt, so he decided the best way to make Alexander happy was to stay away.
And that, of course, didn't work as planned.