 Solution

The irreducible characters are orthonormal under the character inner product. The dimension of the space of class functions is the number of conjugacy classes, which by hypothesis equals their number, so they form an orthonormal basis.

Conjugate elements have equal values under every character. Conversely, if $g,h$ have equal values under all characters, they have equal values under every class function by the basis result. Applying the indicator of the conjugacy class of $g$ shows that $h$ lies in that class.