You can register to file small claims court proceedings online here:
https://www.moneyclaim.gov.uk/web/mcol/welcomeWhen you register, make a note of the 12-digit ID they give you. That's your login. For your address, use the one you created in Step 1. Unless you're in the UK already.. then use your own address

Also make a note of the MCOL Customer number.
Once that's setup, you can Begin your claim (don't click this image, click this on the website you doofus!):

The fee to file is automatically get added to the amount that you're suing Alpha for. But you do have to pay for it upfront using a credit card. Here's the fees:

By the way Alpha's address is:
Alpha Technology (INT) LTD
Manchester Business Park
3000 Aviator Way
Manchester
M22 5TG