Hi
I have done this already (Flew to Bilbao, then bus and train to Bayonne, SJPP, later back from Burgos to Bilbao)
If you haven't been there yet: Biarritz and Bayonne are practically one city.
IMO best connection:
1. Bus from Burgos to Bilbao (roughly every hour, ALSA, 1:50 travel time
www.alsa.es)
2. From Bilbao you can take the next Bus to Irun/Hendaye again ALSA, about 2 hours. To Spanish-French Border. About ten buses a day.
The buses meet in Termibus Terminal in Bilbao. Tickets can be bought at the bus station at the counter, on vending machines there and even online.
Be aware that it is the proud of the bus comanies that the bus starts on time. If the horario says at 14:00 expect it to leave at the exact time.You can easily miss the bus. Ask the staff at the station for the correct platform (="anden")
3. In Irun Hendayae the bus terminates at the railway station. From here take a commuter train to Hendaye station. If you want to walk, it is just about one and a half kilometers.
4.From Hendaye there are regular trains to Bayonne. Even the TVE to Paris starts from here and stops in Biarritz/Bayonne
Railway travel in Spain is possible from Burgos to Bilbao and on to Irun, but this is more exciting and trains are not running as often as busses. There are also different rail systems in spain: Narrow gauge northern network (FEVE), and partially paralell in the north broad gauge (RENFE) as well and the High Speed trains, which partially run on the same tracks as the french railways.
Price Burgos Bilbao ALVIA High speed, two trains a day about 10€, two more regular trains a day about 19€, all nearly 3 hours as running on same track, from Bilbao to Hendaye it is the narrow gauge train.
You would need to take the train to Donostia/San Sebastian the Euskotren Linea 1 from Bilbao Atxurri station to San sebastian. This is not the main Railway station. You could walk it in 30 minutes -45 minutes, or take a taxi. Then the Metro Donostialdea from Donostia to Hendaye/Hendaya.
I love to use the train in spain, but it is a bit more adventurous.
Enjoy your camino!
Here a link to my video diary