...want some sort of human art, and want humans to make it for them. This has a mystery of "why are all these artists disappearing?" (A fairly standard plot, but worth mentioning.)

...think that Earth is a great place to drop their nice little oxy/nitrogen breather nature preserve, which turns out to be an invasive species.

...think that Earth is a great place to dump their *garbage,* which requires an Earthlike atmosphere for some reason -- some sort of biomechanical garbage that falls asleep when exposed to oxygen?

...want something a different invasion (possibly by their enemies) left on Earth?

Yeah, 'X species attacks/invades Earth' is pretty much the bread-and-butter of lots of Doctor Who plots. These can, however, be broken down into a few broad categories:

1. They want to wipe out humanity for TEH EVULZ.

2. They want to use the Earth as a resource for something, and humans are just getting in the way.

3. They want to use the humans as the resources.

4. They just happened to crash-land and ended up getting stranded, and humans mistook it for an invasion.

5. They don't possess the intelligence necessary to realize that what they're doing is wrong, and attack people simply because it's their instinct to find food.

And a puzzle can be anything, from hacking into a computer to figuring out the aliens' motive(s). Puzzles aren't confined to just riddles and physical interlocking pieces, after all. :)

So first, you don't wanna make anything too complex, or try to write the next Sherlock Holmes, especially if it's your first time. And maybe the aliens are invading because they need to take a dump and their toilets are broken. I dunno, I don't watch Doctor who.