Does Kylo not know the fate of Darth Vader?

Question

It seems Kylo has an obsession of Darth Vader and has an interest in the dark side because of it.

Does he not realize that Darth Vader turned good and helped destroy the Empire (Spoilers?)? I feel like Kylo would've seen how much better the Light Side was as it was able to turn Vader from the Dark Side if he had known this.

Answer 1

Kylo Ren knows about Vader's fate, after all,

he trained under the person who was responsible for Vader's return to the Light Side of the force.

Snoke and Ren discuss it in the movie.

From the novelization:

“Yes. Such a simple thing. Such a foolish error of judgment. A momentary lapse in an otherwise exemplary life. Had Lord Vader not succumbed to emotion at the crucial moment— had the father killed the son— the Empire would have prevailed. And there would be no threat of Skywalker’s return today.”

“I am immune to the light,” Ren assured him confidently. “By the grace of your training, I will not be seduced.”

Additionally, JJ Abrams mentions this fact in an interview with IGN

“Kylo Ren idolizes Darth Vader, not Anakin Skywalker,” Abrams explained. “He idolizes what Vader represents and what Vader was trying to do. The idea that Vader didn’t succeed, if you look at it from Ren’s point of view, he was seduced by the enemy and failed because of that seduction. So the idea is that Ren wants to complete the thing that Vader started.”

Answer 2

Kylo Ren knows that Anakin had turned back to the Light. What he knows about Anakin's earlier fall to the Dark Side is unknown as of this time.

Kylo Ren is aware, at the time of The Force Awakens, that Vader had ultimately rejected the Dark Side in favour of the Light — Kylo chooses to view this in negative terms, believing that agents of the Light Side had "seduced" Vader.

In the Alan Dean Foster novelization, Kylo and Supreme Leader Snoke have a conversation in which Snoke says:

"Such a foolish error of judgement. A momentary lapse in an otherwise exemplary life. Had Lord Vader not succumbed to emotion at the crucial moment — had the father killed the son — the Empire would have prevailed."

(Chapter X)

So, if Kylo hadn't known earlier, then he certainly would have known from this point forward that Vader had rejected the path of the Dark Side, at that he did so at least partly because of an emotional response to Luke.

However, Kylo most likely knew about Vader's decision for some time. This is made clear by J.J. Abrams' own explanation of Kylo Ren's motivations:

Kylo Ren idolizes Darth Vader, not Anakin Skywalker. He idolizes what Vader represents and what Vader was trying to do. And the idea that Vader didn't succeed, if you look at it from Ren’s point of view, he was seduced by the enemy and failed because of that seduction. So the idea is that Ren wants to complete the thing that Vader started.

(Source)

Abrams' phrasing makes it clear Kylo Ren's own motivations for following the Dark Side have to do with Vader's rejection of it. Kylo sees this as a wrongdoing committed by "the enemy": Luke and other agents of the Light.

The fact that Kylo Ren is aware of this means that someone — at least one of Luke, Leia, Han, or Snoke — had told him during his upbringing or training.

Finally, what Kylo knows about Anakin's initial fall to the Dark Side — his relationship with Padme, his dealings with Palpatine, his fight with Obi-Wan — is unknown at this time. Nothing is said about this in the film or novelization, and no official statements have been issued.

Answer 3

To complement Praxis's answer:

There is a plausible way that Ben/Kylo Ren may have found out details about Anakin's fall to the Dark Side (Praxis's answer covered his un-falling).

• Bail Organa was with Yoda and Obi-Wan; he took them to Coruscant to find the beacon in Jedi temple and reverse it - where they learned of Anakin's betrayal. Later on he took Yoda away from Coruscant, to the asteroid where Luke and Leia were born, and participated in twins' futures discussions, including their Force training to be.

• While it's not explicitly shown on-screen, it is possible that Yoda and Obi-Wan told Bail Organa what they knew of Anakin's fall - or he guessed parts, what of it due to Padme's state and the rest of info he had from Coruscant.

• And it's not impossible that he imparted that information to Leia, although if he did, she ought to have recognized Luke's Skywalker last name which she didn't. But perhaps he didn't mention Anakin's name to her, till Luke told her that Vader was Anakin.

• If so, it's plausible she would tell the tale to Ben, as a cautionary tale, to prevent him from fulfilling Han's "There's too much Vader in him" mistake.

Another theory would be Luke learning details from Force Ghosts of Yoda and Obi Wan, as they never told him real story (Padme, secret marriage, worry about her life because of what happened to Shmi, friendship with Palpatine, Palpatine telling him that he was Sith, attempted arrest of Palpatine by Windu) in detail while alive; and he'd have told it to Ben for the same precautionary reasons.

Of course, none of this is even remotely canon, merely a not-implausible theory. We simply don't know anything more.