Relation between a representative of a conjugacy class and corresponding irreducible character value Is there a relation between the representative order of a conjugacy class and the corresponding irreducible character value? 
Thanks in advance.

If I understand your question correctly, you are taking an element $x \in G$, and an irreducible character $\chi$, and asking what influence does the order $n = \lvert x \rvert$ of $x$ have on $\chi(x)$.
In this generality, I do not think you can say much more than the fact that $\chi(x)$ is the sum of some $n$-th roots of unity.
This is because, if $\rho$ is a representation corresponding to $\chi$, then $\rho(x)^n = \rho(x^n) = \rho(1) = I$, where $I$ is an appropriate unit matrix, so the minimal polynomial of $\rho(x)$ divides $x^{n} - 1$, and all eigenvalues of $\rho(x)$ are $n$-th roots of unity.
