Diamond unit cell is F.C.C type with four additional C atoms tetrahedrally bonded(black ones). Corner atoms contributes to 1/8 th of the atom to the unit cell, Face centred atoms contributes 1/2 of the atom. Now, Total atom in one cell can be summed up to :

(1/8 X 8) +(1/2 X 6) + 4(tetrahedrally bonded atoms in the cell) = 8.