According to Chinese media reports, over the course of the past two weeks or so, the refitted former Soviet ship has been conducting sea trials with Shenyang J-15 Flying Shark fighters onboard. The J-15 itself is a reverse engineered version of the Sukhoi Su-33 Flanker, which flies off the former Varyag's sister ship Admiral Kuznetsov.
China's rapid progress in developing its naval aviation prowess should not come as a surprise. They've been investing considerable time, effort and money into developing those capabilities. Moreover, they've essentially had a lot of free research and development help from Russia and Ukraine handed to them in the form of the Varyag itself and prototypes of the Su-33. Reverse engineering proven technologies has no doubt greatly accelerated China's progress. Nonetheless, the PLAN's accomplishments can't be dismissed lightly; it's still very impressive.
But Liaoning is but one ship and it still takes time to train up a ship and its air wing to deploy--that's after they complete developmental work and correct any deficiencies they find in test. China also has to integrate the carrier into its fleet and develop a doctrine for its use. The ship will probably be used to develop those concepts of operation and act as a kind of template for future Chinese carriers.
When those future Chinese carriers might start to appear is anyone's guess. There have been some reports that suggest China's first truly indigenous carrier will be launched next year, but so far there is no concrete evidence to suggest further vessels are already under construction... Nor do we know for certain (or at all) if or when the Chinese government might approve more aircraft carriers.