Not a journey I've made, but you can find train times and details in English on the excellent:
Journey time to Bilbao-Abando railway station is just under 6 hours, with 2 changes.
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From Bilbao/Bilbao airport you can get a bus to San Sebastian. Change there for another bus to Biarritz. They are provided by www.pesa.net You could take the train but the Euskotren is much slower. If you have the time it takes a nice route though. Change at Hendaye onto the French SNCF network to Biarritz Gare La Négresse.
InBairritz provided a good alternative. There are regular buses from Bilbao to San Sebastian/Donostia (about 1 hour). From the SS/Donostia bus station there are regular buses up to Biarritz (2 1/2 hours). Check the Pesa site. The Pesa ticket office in SS is just around the corner from where the buses arrive and depart. You can do the train, which switches from the Spanish line to the French line at Hendaye, but I looked at it several times over the past several years and never found it with convenient time schedules.