The raw problem here is simply how big the atmosphere is. Earth's atmosphere masses on the scale of 5 billion billion kilograms. Adding something to it on a planetary scale is much harder than it sounds.
A chemical "chain reaction" would involve transforming the elements already present in the atmosphere into a new configuration. The problem is that the elements in the atmosphere quite like the configuration they're in. Diatomic nitrogen (N2) is generally considered stable and inert, as is argon. Oxygen is highly reactive, but it needs something to react with, and the relatively high levels of oxygen in the atmosphere mean anything susceptible to it at normal temperatures will have been oxidized long before the aliens get involved.
At higher temperatures, oxygen will react to itself (creating ozone) or with nitrogen gas (creating nitric oxide and a variety of related compounds). Both of these are nasty, and quite toxic to humans, although they tend to be short-lived and so presumably wouldn't factor into any kind of permanent alien atmosphere. They also require a lot of energy input: nitrogen combustion occurs in the neighborhood of 2000 °C, and ozone is usually generated by corona discharge. Not something you want unless your alien terraforming (xenoforming?) device is supposed to double as a death ray.
One option would be to introduce a substantial volume of new material into the Earth's atmosphere. The problem, again, is mass. A largish meteor like the Chicxulub impactor might mass somewhere in the vicinity of 10% as much as the atmosphere. But then, that impactor is also thought to have caused the K-T extinction, one of the largest mass extinctions in Earth's history. Spreading out the impacts will reduce the effects on the Earth (although it still won't be much fun), but it will take longer.
The last, probably most practical approach is in situ resource utilization - basically, planting factories on Earth that use materials already present here to change our atmosphere. Of course, that's a long-term solution and rather noticeable for the poor saps already on the planet.
For further reading, you might be interested in ideas for terraforming Mars, essentially the same problem from the alien's point of view.