I'll break this into steps using spoilers so you can try for yourself.
First, calculate what $G=(Z/9Z)^\times$ is.
To do this use the fact that the multiplicative subgroup has order $\phi(9)=6$ where $\phi$ is Euler's totient function, by doing some basic calculations we find the group has the elements $\{1,2,4,5,7,8\}$ and is isomorphic to a cyclic group of order 6.
Now we know what the group is we can apply basic character theory:
We know the group is abelian, so it must have $|G|=6$ conjugacy classes, and thus 6 irreducible representations, as our group is Abelian every representation is linear and thus a character.
We immediately know one of these is the trivial representation and by the representation theory of abelian groups, the other representations are $1$-dimensional and the values take sixth roots of unity.
We can obtain these through lifting from the normal subgroups
For example, we can lift two nontrivial characters from the quotient group of $G/\{1,8\}$ and a non-trivial character from the quotient $G/\{1,4,7\}$.
The full table:
$\begin{array}{c|ccccc}
&1&8&4&7&2&5\\
\hline
ρ_1&1&1&1&1&1&1\\
ρ_2&1&-1&1&1&-1&-1\\
ρ_3&1&1&\zeta_3&\zeta_3&\zeta_3&\zeta_3^2\\
ρ_4&1&-1&\zeta_3^2&\zeta_3&\zeta_6^5&\zeta_6\\
ρ_5&1&1&\zeta_3&\zeta_3^2&\zeta_3^2&\zeta_3\\
ρ_6&1&-1&\zeta_3&\zeta_3^2&\zeta_6&\zeta_6^5\\\end{array}$
where $\zeta_k=e^{\frac{2i\pi}{k}}$.