Note: These instructions do not work with the Nintendo USB Dongle. It is still possible to use the service, but the method depends on your computer's setup (its OS, how the network is configured, etc etc). I may be able to provide generic instructions for this later, but for now either use a "normal" wireless network or find someone who knows how to change the DNS server on your computer.
The first step is to modify the usual Wi-Fi configuration so that GTS will connect to our server instead of Nintendo's.
In your DS's wifi settings (accessible at the bottom of the Continue screen on your Pokemon game),
assuming you've already got a connection to your local wireless network set up, click the settings for that connection.
Scroll down a bit and, where it says something about the "DNS", change Autoconnect to "No",
and manually enter "184.108.40.206" as the primary DNS server.
Save your settings, and go back to the title screen.
Then, go to the Global Trade Station/Global Terminal in Jubilife City (Diamond/Pearl/Platinum) or Goldenrod City (HeartGold/SoulSilver), and connect as usual. After entering and "checking GTS status", the game should immediately send you the Pokémon, and then kick you back to the title screen via a "connection error".
Finally, start your game. Your new Pokémon should be present, either in your team or in Bebe/Bill's PC!
Don't forget to put your wifi settings back to how they were before if you want to use official wi-fi stuff ;)