Firstly I suggest going to www.alsa.es and put in your requirements. This website is is very straight-forward and easy to use.
It takes approx 7 hours and 15 minutes to get between Alicante and Barcelona and costs between 40 and 46 euros - one way (correct as of July 2009, this is subject to change).
There are about 12 buses a day to Alicante from Barcelona and the first bus leaves Barcelona at 6.15 and the last bus is at 23.59
For the return journey of Alicante to Barcelona the first bus leaves at 00.25 and the final bus at 22.05.
If you are planning on taking the ALSA bus then please do check the timetables on their website for up to date information.
Further information on Alicante Buses
Alicante to Benidorm Bus
Alicante to Barcelona Bus
Alicante to Javea Bus
Alicante to Valencia Bus
Alicante to Granada Bus
Alicante to Torrevieja Bus
Alicante to Denia Bus
Alicante to Madrid Bus
Alicante to Almeria Bus
Alicante to Altea Bus
Alicante to Murcia Bus
Alicante to Calpe Bus