How would I go about setting up my own repeater in the UK? For instance, what license category would I need, what equipment would should be looking to purchase, and do I have to register it anywhere?
Repeaters take time and money to set up and maintain. Many repeaters are operated by a club or a group - you may want to consider pooling resources with other local hams.
Unlike in the US, UK amateur repeaters are individually licensed. You must get approval from Ofcom, including a Notice of Variation. This is because your base license does not allow repeater operation, the Notice of Variation modifies the terms of your license. There are both technical and regulatory requirements, including yearly notifications to Ofcom that the repeater is still running.
Before you do so, you'll need to find a frequency that isn't already used in your area. The frequency coordination is done through ETCC.
Working with someone in your area directly is the best way to start this process. Don't try to dive in to all the forms until you've been able to work out your plan - including a site and the actual hardware - especially if this is your first repeater project.