I took our dog to the UK many many years back. ( and then brought her back with us a few years later) I did all the running around and paperwork as vidtek has told you about. I bought the dogtainer through jet pets, and rang all the government departments and vets ( vets are extremely helpful in getting all the right jabs/health checks etc.) Although there is no quarantine in the UK we had to get our dog Rabies injections, so that took 6 months ( we actually started the process, went to the uk, found jobs and somewhere to live and then I flew back and go her....the money I saved from doing it all myself is what paid for my 4 week trip to pick her up)
the biggest cost is the airline ticket, which was then, roughly 2 x a normal adult one way ticket. The price is based on the weight of the animal. Our dog was 21kg and she cost just over $2000. The vet and paperwork added up to around $800 all up. I imagine cats would be a lot cheaper.
Now , it is pretty straightforward, but there is ALOT of paperwork , lots of running around, calling departments etc. At one point I got so utterly sick of AQUIS asking me why I wasn't just using a pet transport company I screamed at them. They really hate dealing with the ' general public' but I did it and she got there. It was much easier bringing her back, but then she had 6 weeks in Quarantine which cost a fortune.
Any way, good luck with it. It can be done by yourself if you want to save money. But prepare for some head aches.