There are lots of these kind of stories, especially during the early Nicieza & Lobdell runs, for better or worse. But ignoring those for a bit, here are some of my favorites:
For Storm's dealing with love and loss, go for the classic LifeDeath (Uncanny X-Men #186):
Uncanny X-Men #182, for a great look on the whole Rogue/Carol Danvers tragedy and how it affects Rogue's mental health:
For some of the best interactions and overflow of our heroes' emotions, the iconic X-Men #70:
There are MANY MORE fantastic character-driven stories in the franchise, including the spin-offs: New Mutants, X-Factor, X-Force, Generation X, New X-Men.
X-Men Legacy provides some of the best written character-driven stories, courtesy of Carey. The focus is Professor X and it's so, so good.