Bath, I don't mind travelling though

I feel like I have to jump in here in regards to black ink fading. I have a friend who is a very talented, professional tattoo artist (he's done one for me as well as several of our friends, and they're all still vibrant with sharp, crisp lines years later), and I can tell you truthfully that black doesn't always fade. However, many factors determine the quality of a tattoo as it ages.

There are really three main things that are important for a good tattoo: the type of ink, the tattoo technique, and aftercare. Cheaper inks can fade quickly, many times within a matter of weeks. As for technique, if the tattoo artist (or technician in your case) is too heavy-handed it can cause much of the ink to bleed out (diuretics in your system can cause this, too, so it's best to avoid alcohol, caffeine, etc. before getting a tattoo). Being too light handed can give the tattoo a fuzzy appearance rather than a sharp, crisp line.

Equally important is how a tattoo is cared for after getting it. The followup routine generally recommended by tattoo artists involves keeping the tattoo moisturized with some type of ointment the first few days afterward and cleaning the area with antibacterial soap (I'm sure for permanent eyeliner you would have to use specific ophthalmic products for this). The tattooed area will itch, and the skin will peel and possibly scab; tattoo artists always stress not to pick at it, no matter what because it can make a huge difference in how the tattoo looks after it heals. Once it heals (which usually takes a few weeks), it's also important to avoid sun exposure. Most people either keep it covered with clothing or with a high SPF sunscreen, but obviously that's not possible for a tattoo in the eye area, so you'd probably need to always wear a wide brim hat and polarized sunglasses when in the sun.

***Now with all of that being said, even though it's possible to have a black tattoo that doesn't fade, my personal opinion is that you shouldn't go the permanent eyeliner route. Despite permanent makeup being done in a medical setting (rather than by a tattoo artist in a shop), the number of possible complications makes me wary because a complication could affect your vision. As Mo said, that is not an area to take risks with!

I would really go consult with the best tattooist in your area about all these things, then see if they recommend a technician. I don't know about the UK, but over here there are some pretty lazy regulations about permanent make-up, and it gets fairly dicey.