This scene really defines Snape, I think. Anything he does outside of showing hatred towards Harry prior to the last chapters of the last book is purely on Dumbledore’s orders.
My theory is that Dumbledore would have discussed with Snape privately after he’d received yet more feedback from each of his staff regarding how Harry was doing. (forgive me on this part as I don’t have the appropriate citing) One of the teachers passed information to Dumbledore than he was rather ill-equipped at even the most basic means of defending himself. After much consideration, Dumbledore requested Snape display an open invitation, to any willing students who wished to lean how to defend themselves. Snape has agreed to do anything Dumbledore requests of him.

It was not Lockhart that decided to start up the Duelling Club – we know this because he’s an almost completely untalented wizard. Although he invited himself up to Dumbledore’s office to offer a suitable detention (signing fanmail) when he overheard only the last part of the conversation, and true to character offered to help with the extra-curricular lessons; he did, after all, defeat [some Dark wizard], [whatever year that didn't actually happen].

And so it was agreed. Snape and Lockhart were to teach extra-curricular duelling, much to Snape’s dislike.

Skip forward to the actual duelling.

Snape’s first move on Lockhart is the Expelliarmus charm, although the effect is much more than just the wand flying out of Lockhart’s hand. He’s blasted backwards. My assumption is that he used a non-verbal spell in addition to the verbal “expelliarmus”.

Onto my last note.

I have my suspicions regarding Snape’s deliberately outing Harry as a Parselmouth.
I think there was an unseen memory from the set of memories that night Snape was killed (DH). When Dumbledore told Snape that part of himself had lodged itself into Harry, Snape didn’t want to believe it. But over the course of the first year, and part of the second, he had come to truly loathe Harry for being so physically close to James, that his emotions grew slightly detached from him. It took him more than 12 months to test out Dumbledore’s theory – and what better a method than to test it out where he was under pressure and had to act instinctively?

2. Why was this the only time there was a dueling club?

And yes, I don’t think Dumbledore has had anything to do with this dueling charade. Like what the others said, he doesn’t seem like the type to openly expose Harry’s connection with Voldemort like that. And yes, I agree with what Casey said: that Dumbledore simply told Snape to “look into it” and Snape decided to find out using Serpensortia. It’s actually killing two birds with one stone: by being in the duel, he could help protect Harry if anything goes wrong (like what John said) and he could find an answer to his question.

This is just my assumption, though. I think it sort of fits the personality of the characters involved and the underlying plot of the book.

“I know,” said Harry. “I mean, that’s only the second time I’ve ever done it. I accidentally set a boa constrictor on my cousin Dudley at the zoo once – long story – but it was telling me it had never seen Brazil and I sort of set it free without meaning to. That was before I knew I was a wizard…” “A boa constrictor told you it had never seen Brazil?” Ron repeated faintly.

I just love reading it out loud to myself. I can just picture how he says it. Quite quickly but like it’s no big deal and with a half roll of his eyes at “long story”. I love picturing what Ron and Hermione’s faces are like as he’s talking. It’s one of the few parts that I read to myself in the manner of the characters.