This is a very hard question to answer. It could be only a few if you define knowing that it existed as actually having been there--or it could be a lot more, who were told that it had been opened or heard it as a story.
Let's begin with those who knew it existed, because they were there, before Book 2.
Salazar Slytherin, but not the other three.
The story goes that Slytherin had built a hidden chamber in the castle, of which the other founders knew nothing.
He opened it in his fifth year, and again in Book 2, through Ginny.
The basilisk did live in the Chamber, and therefore I assume knew about its existence.
So those are the people who've been in the Chamber. But wait...there's more!
Aragog and Hagrid
Did Aragog enter the Chamber of Secrets? I don't think so; but he was blamed as the monster and Hagrid as the opener, so they presumably knew that it existed.
Here's something Tom Riddle says to Harry in his diary:
In my fifth year, the Chamber was opened and the monster attacked several students, finally killing one. I caught the person who’d opened the Chamber and he was expelled. But the headmaster, Professor Dippet, ashamed that such a thing had happened at Hogwarts, forbade me to tell the truth. A story was given out that the girl had died in a freak accident. They gave me a nice, shiny, engraved trophy for my trouble and warned me to keep my mouth shut.
That explains why it's not public, but there are more people who knew about it.
Armando Dippet and Hogwarts' faculty, including Dumbledore
They must've known about the Chamber; Dippet was the one who began to conceal it.
Here's a quote from the wiki:
Aragog was framed as being the monster and Hagrid was accused of being Slytherin's heir.
You can't frame something like this without an audience; someone must've known about this, and news travels fast among these kids. They may not have known, but they would have suspected. Count them if you want to.
Did Lucius know?
“So — have you stopped the attacks yet?” he sneered. “Have you caught the culprit?”
“We have,” said Dumbledore, with a smile.
“Well?” said Mr. Malfoy sharply. “Who is it?”
“The same person as last time, Lucius,” said Dumbledore. “But
this time, Lord Voldemort was acting through somebody else. By means of this diary.”
From this, I doubt that Voldemort would tell him his plan, but he probably knew about it from the rumors.
EDIT: found this quote. It's not proven, and it's only one of Dumbledore's shrewd ideas. On the other hand, those usually turn out to be correct:
I understand that Voldemort had told him[Lucius] the diary would cause the Chamber of Secrets to reopen, because it was cleverly enchanted.
Did Myrtle know?
“How?” said Harry.
“No idea,” said Myrtle in hushed tones. “I just remember seeing a pair of great, big, yellow eyes.
So no, she did not know that it existed, she just died. She might be counted among those Hogwarts students who heard the rumors.