J.K. Rowling reveals Lupin’s history from birth to death in a lengthy new piece within the final Prisoner of Azkaban chapters on Pottermore. This includes, of course, how Tonks and Lupin met.
The two were first introduced to each other in The Order of the Phoenix.
The new group “included and Auror who had been too young to belong to the Order during its first incarnation. Clever, brave and funny, pink-haired Nymphadora Tonks was a protégée of Alastor ‘Mad-Eye’ Moody, the toughest and most grizzled Auror of them all”.
Rowling explains Lupin’s first impressions of Tonks:
“Remus, so often melancholy and lonely, was first amused, then impressed, then seriously smitten by the young witch. He had never fallen in love before. If it had happened in peacetime, Remus would have simply taken himself off to a new place and a new job, so that he did not have to endure the pain of watching Tonks fall in love with a handsome, young wizard in the Auror office, which was what he expected to happen. However, this was war; they were both needed in the Order of the Phoenix, and nobody knew what the next day would bring. Remus felt justified in remaining exactly where he was, keeping his feelings to himself but secretly rejoicing every time somebody paired him with Tonks on some overnight mission.”
Remus had never been in love, and Tonks had a huge impact on him:
"It had never occurred to Remus that Tonks could return his feelings because he had become so used to considering himself unclean and unworthy. One night when they lay in hiding outside a known Death Eater’s house, after a year of increasingly warm friendship, Tonks made an idle remark about one of their fellow Order members (‘He’s still handsome, isn’t he, even after Azkaban?’). Before he could stop himself, Remus had replied bitterly that he supposed she had fallen for his old friend (‘He always got the women.’). At this, Tonks became suddenly angry. ‘You’d know perfectly well who I’ve fallen for, if you weren’t too busy feeling sorry for yourself to notice.’
Rowling explains that “Remus’s immediate response was a happiness he had never experienced in his life, but this was extinguished almost at once by a sense of crushing duty. He had always known that he could not marry and run the risk of passing on his painful, shameful condition”.
“He therefore pretended not to understand Tonks, which did not fool her at all. Wiser than Remus, she was sure that he loved her, but that he was refusing to admit it out of mistaken nobility. However, he avoided any further excursions with her, barely talked to her, and started volunteering for the most dangerous missions. Tonks became desperately unhappy, convinced not only that the man she loved would never willingly spend time with her again, but also that he might walk to his death rather than admit his feelings.”
We also learn from this entry that Lupin’s patronus is an ordinary wolf, not a werewolf.
Rowling says that “Remus dislikes the form of his patronus, which is a constant reminder of his affliction. Everything wolfish disgusts him, and he often produces a non-corporeal patronus deliberately, especially when others are watching”.
These details, along with many more, can be found here in Pottermore.