This tool is under development, so I wouldn't use it personally.
Step 1. Set your primary DNS server to the address at the top of the page in the WFC settings for your DS. Change the settings (advanced settings if using WPA connection) and hit "Save Settings".
Step 2. Make a deposit request for the Pokémon you want to check. When asked for what Pokémon you're looking for, enter anything. When you confirm the offer, the game will tell you your Pokémon can't be traded (but it has been recorded on Pokécheck). If for any reason your Pokémon is accepted, you are on the official GTS and you didn't follow step one correctly, make sure you're setting up the right connection (check advanced settings if using WPA).
You should get error code 13266.
Step 3. Check back to the page to see the stats of your Pokémon. You must connect with the same connection (i.e. IP address) as your DS for the script (so the DNS IP) to find which Pokémon you sent.
If your Pokémon doesn't show up try accessing Pokécheck directly from your DS using the built-in DSi browser, or create a Pokécheck account and see how to register your save file on the page.
Once that's all said and done, search for your Pokemon, OT or Pokemon ID and find your Pokemon at the list. Say I deposited a Scrafty as "Kenny" with the trainer ID 14529. I would search it up, and then I would come upon its summary.
Once there, I go to the bottom of the page and click "Show Legality Analysis". Here's what it came up with:
Green is legal, yellow is iffy, and red is definitely hacked. Since there are no red marks, this Scrafty is legitimately bred.
After checking, just take it back from the GTS. This is how Pokecheck works and is used.
Source: Testing it out myself and prodding around the page