Please tell me if the sentence is correct:
Not only does he speak English well but also is good at French.
Is it necessary to add "he" to the sentence to help it become a good one?
Unfortunately, the sentence is incorrect. If you begin it with "not only," you will need to insert "he" after but and move "is" so that it is between "is" and "good":
(A) Not only does he speak English well, but he is also good at French.
(Notice that I have also added a comma between the two clauses.) If you want to use "but is also good" without "he," you will need to adjust the beginning:
(B) He not only speaks English well but is also good at French.