How do I import the private key onto my client?
I didn't give you a private key, I gave you a partprivkey that was generated from your public key.
Step 4: Once you receive our private key part back move onto step 2 Calculate your vanity wallet In the first field paste the private key you saved into it. Then in the second field paste in our part of the private key. Make sure you click Add then Calculate Vanity Wallet
Step 5: Copy and paste the Vanity Private Key (WIF) into your preferred wallet.
Step 6: Celebrate!!

That's just it though I wasn't given a Vanity Private Key (WIF)
This is the message I receive:
Vanity Private Key (WIF):
Only available when combining two private keysThe only thing I'm given is the Hex version of my Public Key.