You are given a specific hero for clearing certain chapters in the story mode.

Servant	How to unlock
Archer - David	Clear Third Singularity - Oceanus.
Assassin - Henry Jekyll & Hyde	Clear Fourth Singularity - London.
Berserker - Kiyohime	Clear First Singularity - Orleans.
Caster - Cu Chulainn	Clear Singularity F - Fuyuki.
Caster - Geronimo	Clear Fifth Singularity- E Pluribus Unum.
Rider - Boudica	Clear Second Singularity - Septem.