Almost all budget airlines from UK and Ireland fly to Faro airport as well as other airlines flying from other European countries. To get to Alvor, from Faro airport, you may hire a car or take a taxi.
If you hire a car, you have to take the direction Portimão/Lagos and drive through Via do Infante (highway). You leave the highway at Alvor exit and follow the signs to the centre of Alvor and then to the beach.