I was wondering if there is a characterisation for $C$*-algebras (unital) for which the bidual does not have any central atoms.

It is not sufficient for example to demand that the $C$*-algebra does not have a non-zero commutative algebra direct summand: 

Consider the $C$*-algebra $$A:=\bigl\{f\in C([0,2];\mathrm{M}_2(\mathbb{C}))\colon \ f \mbox{ is diagonal on } [0,1]\bigr\}.$$

Then $A$ does not have a non-zero commutative algebra direct summand, but the atomic part of the bidual should be 
$$A^{**}_{atomic}=\bigl\{f\in\ell^\infty([0,2];\mathrm{M}_2(\mathbb{C}))\colon f \mbox{ is diagonal on } [0,1]\bigr\}$$

which has plenty of central atoms. See https://mathoverflow.net/questions/313540/commutative-direct-summands-of-c-algebras.

It is true that the bidual of a $C$*-algebra $A$ has no central atoms if and only if the state space $S(A)$ of $A$ has no singleton split faces. This is [Corollary 5.33 and Corollary 5.34](https://books.google.com/books/about/Geometry_of_State_Spaces_of_Operator_Alg.html?id=YBPSBwAAQBAJ&pg=PA162) in *Geometry of State Spaces of Operator Algebras* by Alfsen and Shultz. This characterises these algebras in terms of the state space, but knowing a condition on $A$ would be very interesting.