How/When did Liz learn about Kirito's Duel Wield?
In the novel, after Lisbeth made the sword (and before Asuna barged in), she did ask him what he needed it for:

"Speaking of which— In the beginning, you did say, a sword equal to this one, didn't you. That white one certainly is a good sword, but I don't think there's much difference with that monster dropped one. Why would you need two swords resembling each other?"
"Aah..."
Kirito turned about, staring at me with an expression that showed that he was hesitating over something.
"Well, I can't explain everything. If you won't ask any further than that, I can tell you."
....

And then he showed his skill to Lisbeth. The anime skips this entirely.