If you want to be fully legal in owning a scooter, just buy one. End of the story.
If you want to be fully legal when you ride it though here's what you need:
- license for 2 wheeled vehicles (there are different licenses depending on the engine volume).
- If you want to drive inside the 4th ring road you need a "京A" plate on your scooter. The beijing government stopped issuing such plates or at least slowed down a lot, so you need to buy one second hand, and have the certificate that this plate's former vehicle
has been destroyed. Last time I checked, prices for such a plate were in the likes of 8 to 10 thousand RMB.
Some say that a plate from another province is ok too, which would be a LOT cheaper. Have no idea if this is true or not.
Once you get the plate, you need to register it for your new vehicle.
Then I guess that's it you should be able to legally drive your scooter in Beijing!