There are 38 Data Core Doctor Who cards in total, and 30 are unlocked one by one when you complete each stage of the Adventure Game. There are 8 other cards you are able to unlock by playing the mini games.

Card	How to Unlock
Auton and Tallulah	Score at least 25000 points in Cyber Sequence
Judoon Captain and Brannigan	Score at least 25000 points in Interrogation
Novice Hame and Sisters of Plenitude	Score at least 25000 points in Memory Matrix
Sycorax and Ood	Score at least 25000 points in Hi/Lo