12

It is indicated that he already created the ring Horcrux by the time he approaches Slughorn, it is, presumably, later than he discovers basilisk and kills Myrtle. Though there is no clear indication whether he knew who his mother and her family were by the time he created diary Horcrux.

On the other hand, he found the COS, basilisk and his connection before he created the diary, as his soul piece in the diary states that he is the descendant of Slytherin. So there are 2 possibilities: he first found out about his ancestors and then opened the Chamber or he somehow managed to find the Chamber and afterwards concluded his origins by his ability to control the basilisk.

So, did he learn about his connection to the Gaunt family before he discovered the COS or after?

14

He knew his ancestry before making Horcruxes.

The Dark Lord knew his ancestry before he made his first Horcrux. He began his search by looking for his father in wizarding records, but was unable to find any evidence of him in them.

“It seems that he searched in vain for some trace of Tom Riddle Senior on the shields in the trophy room, on the lists of prefects in the old school records, even in the books of wizarding history. Finally he was forced to accept that his father had never set foot in Hogwarts. I believe that it was then that he dropped the name for ever, assumed the identity of Lord Voldemort, and began his investigations into his previously despised mother’s family – the woman whom, you will remember, he had thought could not be a witch if she had succumbed to the shameful human weakness of death.”
- Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)

When he was sixteen, he discovered the Gaunt line, and that he was a descendant of Salazar Slytherin. When he went to find the Gaunts, he knew they were his relatives - that’s why he sought them out.

“All he had to go upon was the single name “Marvolo”, which he knew from those who ran the orphanage had been his mother’s father’s name. Finally, after painstaking research through old books of wizarding families, he discovered the existence of Slytherin’s surviving line. In the summer of his sixteenth year, he left the orphanage to which he returned annually and set off to find his Gaunt relatives. And now, Harry, if you will stand …”
- Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)

Whether he made the diary or the ring into a Horcrux first, he would’ve known his ancestry first. He made the diary into a Horcrux because it was proof he was he Heir of Slytherin, so he had to have known first.

“Lord Voldemort liked to collect trophies, and he preferred objects with a powerful magical history. His pride, his belief in his own superiority, his determination to carve for himself a startling place in magical history; these things suggest to me that Voldemort would have chosen his Horcruxes with some care, favouring objects worthy of the honour.’

‘The diary wasn’t that special.’

‘The diary, as you have said yourself, was proof that he was the heir of Slytherin; I am sure that Voldemort considered it of stupendous importance.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)

When he comes out of the diary, he does somewhat imply that he knew about the existence of the Chamber for five years before opening it, but he may be obliquely referring to discovering his ancestry and then through that realizing he could open the Chamber instead.

“But I admit, even I was surprised how well the plan worked. I thought someone must realise that Hagrid couldn’t possibly be the heir of Slytherin. It had taken me five whole years to find out everything I could about the Chamber of Secrets and discover the secret entrance … as though Hagrid had the brains, or the power!”
- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)

He only took the ring after going to find the Gaunts, and when he found Morfin, he soon took his wand and used it to kill the Riddles. He did this because he wanted to kill the Muggle who abandoned his mother.

“Voldemort Stupefied his uncle, took his wand, and proceeded across the valley to “the big house over the way”. There he murdered the Muggle man who had abandoned his witch mother, and, for good measure, his Muggle grandparents, thus obliterating the last of the unworthy Riddle line and revenging himself upon the father who never wanted him. Then he returned to the Gaunt hovel, performed the complex bit of magic that would implant a false memory in his uncle’s mind, laid Morfin’s wand beside its unconscious owner, pocketed the ancient ring he wore and departed.”
- Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)

At that point, he knew his entire history - he already knew both that he was related to the Gaunts and through them to Salazar Slytherin, and he knew Tom Riddle Sr. was a Muggle.

3

Yes he knew about his ancestry before opening the chamber of secrets and starting his horcruxes business.

Learning about his ancestry was his obsession as soon a he joined Hogwarts and it seems it is what started his whole arc, because he became upset being not pure-blood while at the same time being a descendant of Salazar. His parents' story (him being half blood, never receiving love from his dead mother or absent father, his mother being a kind of "failed" wizard according to her family) is the cause of his story (becoming the bad guy).

Here is a quote from http://harrypotter.wikia.com/wiki/Tom_Riddle boldface from me

Upon arriving at Hogwarts, Riddle became obsessed with his heritage and began researching it with an insatiable hunger. He focused solely on the identity of his father, thinking him to be the magical parent, as he felt his mother could not have been a witch if she had died. He searched for his father's name in the school trophy room, in the records of Hogwarts prefects and in records of wizarding history, but found nothing to suggest his father had even attended Hogwarts. He was eventually forced to accept that his father was the Muggle parent, and that his mother was the magical one. It was around this time that Riddle gave himself the alias "Lord Voldemort", to spare himself of the reminder of his "filthy Muggle father."[18]

Using his middle name, Marvolo, which was taken from his maternal grandfather, Riddle discovered his mother's heritage and the Gaunts' bloodline connection with Salazar Slytherin. Once Riddle learned of his ancestry, it was not long before he discovered the existence of the Chamber of Secrets under Hogwarts, during his fifth year, and tamed the Basilisk which dwelt within.

Still from this biography page, we can read that it was later after these events that he went back to kill his paternal family and steal the Gaunt ring.

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