Steven's and Gjergji answers points that there is no bijection, however possibly this idea should not be put into the rubbish completely.
Ideologically conjugacy classes and irreducible representations are somewhat dual to each other.
The other instances of this "duality" is Kirillov's orbit method - this is "infinitesimal version" of the duality: orbits in Lie algebra are infinitesimal versions of the conjugacy classes. But pay attention orbits are taken not in Lie algebra, but in the dual space g^. This again manifests that there irreps and conj. classes are dual to each other. However think of semi-simple Lie algebra - then g^ and g can be canonically identified...
Another instance is Langlands parametrization of the unitary irreducible representations of the real Lie group G. They are parametrized by conjugacy classes in Langlands dual group G^L. Again here are conjugacy classes in G^L, not in G itself. However for example GL=GL^L...
So it might be one should ask the question what are the groups such that conjugacy classes and irreps are in some natural bijection or something like this ?
PS
Here is some natural map conjugacy classes -> representations. But it does not maps to irreducible ones, and far from being bijection in general.
A colleague of mine suggested the following - take vector space of functions on a group which are equal to zero everywhere except given conjugacy class "C". We can act on these functions by $f \to g f g^{-1} $ - such action will preserve this class. So we get some representation. In the case of abelian group this gives trivial representation, however in general, it might be non-trivial. It always has trivial component - the function which is constant on "C".
I have not thought yet how this representation can be further decomposed, may be it is well-known ?
