Before you take the route of re-registering your English car, you need to decide if you are happy drive on the right-hand side of the road with a left hand drive car. May people do it, but at junctions, roundabouts, motorways etc. you are always in the wrong position, which can be frustrating and difficult in certain situations.
It is possible to re-register your car in person, but most people will hire a Gestor. You have to make sure your headlights are correct, in some cars you are able to switch the headlights to change directions, but most of the time you will have to replace the headlights. It requires a visit to Trafico in Alicante, an ITV test, which is similar to an MOT test and then you have to pay an import tax related to the type of car and its CO2 commissions. A rough ball park figure to do all the above via a Gestor costs around 1000 euros. A Gestor will be able to give you a quote on how much it will be for a specific car.