The first thing you need to know is that "leaves" means pages of a book, not leaves on a tree. "Lines" means lines of a poem. And you know what "rhymes" means. The poet is saying that the pages on which his poem is written are lucky, since the hands of the woman he loves will be touching them. The lines he has written are lucky because her eyes will be focused on them. His rhymes are lucky that she will read them. (Mount Helicon is important in Greek mythology because it is sacred to the Muses who inspire poetry and other arts. Look it up for more information.) If his beloved reads and enjoys his poem, he doesn't care what anybody else thinks of it. That's what " I care for other none" means.
One note on the old-fashioned language of the poem. At the end of line 2, "might" means "strength" or "power." So "dead doing might" means "power to kill," or "life-and-death power." If there are other words or phrases you don't understand, look them up.