A It was only after Katharine Graham became publisher of The Washington Post in 1963 that it moved into the first rank of American newspapers, and it was under her command that the paper won high praise for its unrelenting reporting of the Watergate scandal.
This is wordy and lots of "it" but nothing wrong, so keep it for time being.
B. It was only after Katharin Graham's becoming publisher of The Wasington Post in 1963 that it moved into the first rank of American newspaper, and under her commandt it had won high praise
Incorrect as this sentence does not tells us about Katharine Graham (it tells us about Katharine Graham's), so her has no referent.
C. Katharine Graham became publisher of The Washington Post in 1963, and only after that did it move into the first rank of American newspapers, having won high praise under her command
Not clear who won high praise under her command. She herself, Washington Post, or American newspapers. Finally "having won high praise under her command" leaves a sentence fragment.
D. Moving into the first rank of American newspaper only after Katharine Graham became its publisher in 1963, The Washington Post, winning high praise under her command
This sentence makes "The Washington Post" optional item. Moreover it also creates a run on sentence, and using wrong verb tense as well.
E. Moving into the first rank of American newspapers only after Katharine Graham's becoming its publisher in 1963, The Washington Post won high praise under her command.
Incorrect as this sentence does not tells us about Katharine Graham (it tells us about Katharine Graham's), so her has no referent.
So remaining one is A.