I was wondering: Exactly why does Tamaki's grandmother have a deep dislike of him?

Although there's no way to be sure, I would guess that she dislikes him for being illegitimate, and therefore disgracing the family （and of course that's not even remotely his fault）. And beyond that point, she must have blamed Tamaki further for making his existence known rather than staying in France, and proceeding to inherent the company, something an illegitimate son doesn't seem to "deserve". So essentially, his grandmother resents him for being the स्रोत of a स्कैंडल and regrets that he is not treated as exactly that.

If आप watch the 25th या the 26th episode then आप realize it is because of Tamaki's mother I believe is why Tamaki's grandmother hates him