Hire a car if you are a driver.
'...cannot seem to fly direct': Yup. Only a few direct flights in the summer months from Madrid on Air Nostrum on F/Sa/Sun or on Tu/W/Sa ex Barcelona with Vueling. Expensive as well when you add the cost of getting to Madrid or Bacelona.
Bus: A long bus journey from Murcia city centre bus station. You'll need to change buses in Sevilla. It's about 7-8 hours to Sevilla from Murcia city centre and then another 3-ish hours from Sevilla to Albufeira. Plus waiting around time for connecting buses. You can easily and cheaply get a bus to Murcia city centre bus station from San Javier/la Ribera.
Bus info if you wish to consider that option: http://www.movelia.es/es/index.html (in multiple languages).
There may of course be other options that I'm unaware of.
You could always do the Spain component part of the rental from Murcia-San Javier (MJV) → Sevilla. There are several car hire companies located around Sevilla's Santa Justa train station that also operate out of MJV...Europcar, Centauro, Enterprise amongst 'others'.
You'd obviously then need to go through the car hire process in Albufeira again.