Hi Tony
We've not used Murcia airport yet but have driven from Alicante airport to CDA and it took about 1 hour 15 minute, so here are the directions.
As you leave Alicante airport follow signs towards Valencia/Murcia
At the A7 junction veer to the left signposted Murcia
You stay on the A7 all the way to Murcia and then take junction 138 which is still the A7 signposted Almeria/Granada
Stay on this road until junction 627, direction Cartagena/Mazarron and as you come off the slip road take junction 627B
Shortly after this take Junction 4 direction Mazarron
At the roundabout turn right on to RM23 towards Mazarron
Take Junction 5 towards Condado de Alhama
At roundabout turn left up the road with palm trees
At next roundabout take 2nd turning towards “Bulevar Central de las Canadas”
Go straight over next 2 roundabouts and at the 3rd roundabout turn right towards “Puerta Las Canadas Sur”
Here you will go through security barriers and need to show ID.
The only problem we had was when we left the A7 for junction 627 we took A instead of B so just be carefully at that point otherwise it was easy enough to find CDA. Obviously on the return leg we did not have that problem and had trouble getting to Alicante airport.
Good Luck
Bee