As you leave Alicante airport follow the sign for A31 direction Madrid and join A7 direction Valencia, Benidorm.
Take exit 71B from A7 to join A31 towards Madrid
On A31 take exit signposted for Aspe, Elche, Murcia
Keep following the sign for Aspe until you pass an Agip petrol station (yellow canvas) on your left. Take a right turn signposted for Alenda Golf (campo de golf).
Follow the road up the hill until you see four appartment blocks on your left
This is a link to a useful route planner. This should show you the route from Alicante airport to Casas de Alenda, which is the entrance to the golf complex.
Several budget airlines fly to Alicante airport :
Easyjet : Luton, Bristol, East MIdlands, Edinburgh, Glasgow, Liverpool, Gatwick, Stansted, Newcastle
Monarch Airlines: Luton, Gatwick, Birmingham, Manchester
ThomsonFly : Gatwick, Manchester, Newcastle, East Midlands, Doncaster, Sheffield, Coventry, Cardiff, Bournemouth
FlyBe : Exeter, Norwich, Southampton
BmiBaby : Heathrow, Manchester, Birmingham, Cardiff, East Midlands
British Midland : Aberdeen, Belfast, BIrmingham, Cardiff, East Midlands, Edinburgh, Glasgow, Leeds Bradford, Heathrow, Manchester
Jet2.com : Leeds Bradford, Manchester, Blackpool
FlyGlobeSpan: Aberdeen, Durham, Edinburgh, Glasgow
Ryanair : Liverpool, Stansted
GB Airways : Gatwick
Also worth checking British Airways as they sometimes have special offers.