Expanding slightly on the other answers:

To ask for a "natural" bijection is presumably to ask for a natural isomorphism between two functors from the category of finite groups to the category of sets. First, we have the contravariant functor $S$ that associates to each $G$ the set of isomorphism classes of irreducible representations. Then we have the covariant "functor" $T$ that associates to each $G$ the set of its conjugacy classes.

The first problem is that $T$ is not in fact functorial, because the image of a conjugacy class might not be a conjugacy class. So at the very least we should restrict to some subcategory on which $T$ is functorial, e.g. finite groups and surjective morphisms.

But the key problem still remains: There is no good way to define a natural transfomation between two functors of opposite variances. So when I said in my earlier answer that "I do not see any way you can make this natural" I might better have said "This is not a situation in which the notion of naturality makes sense".

All of this, of course, is really just an expansion of Gjergji's example.

Expanding slightly on the other answers:

To ask for a "natural" bijection is presumably to ask for a natural isomorphism between two functors from the category of finite groups to the category of sets. First, we have the contravariant functor $S$ that associates to each $G$ the set of isomorphism classes of irreducible representations. Then we have the covariant "functor" $T$ that associates to each $G$ the set of its conjugacy classes.

The first problem is that $T$ is not in fact functorial, because the image of a conjugacy class might not be a conjugacy class. So at the very least we should restrict to some subcategory on which $T$ is functorial, e.g. finite groups and surjective morphisms.

But the key problem still remains: There is no good way to define a natural transfomation between two functors of opposite variances. So when I said in my earlier answer that "I do not see any way you can make this natural" I might better have said "This is not a situation in which the notion of naturality makes sense".

All of this, of course, is really just an expansion of Gjergji's and Qiaochu's observations.