This is a question on a Banach algebra setting.Let $X$ be a locally compact Hausdorff space.
Consider $C_0(X)=\{f:X\longrightarrow \mathbb{C} $, continuous :for every $r>0$ there exists $K_r\subseteq X$ compact with $|f(x)|\leq r$ for $x \in X\setminus K_r \}$
We know that this is a banach algebra with the sup norm and pointwise operations and the point is to identify the characters of this Banach Algebra.
My attempt is to consider a functional $\phi $ on the dual space which is identified by a finite total variation measure $m$ so as $\phi(f)=\int_{X}fdm$. Now the multiplicativity (if that's a word) gives $ \phi(f^2)=\int_{X}f^2dm=(\int_{X}fdm)^2$ or $\int_{X}(f^2-(\int f dm)f)dm=0$...(now i need help). I suspect that this functional should look like the fourier coefficient of the function multiplied by something...
The dual of $C_0(X)$, where $X$ is locally compact Hausdorff, is indeed the isomorphic to the space of regular Borel measures $\mu$ on $X$, where $\mu$ defines a functional $f \to \int fd\mu$ and all functionals are of this form. See wikipedia, e.g.
A character is stronger, namely it's multiplicative. All non-zero ones of those correspond to all point-measures $\mu_p(A) = 1$ iff $p \in A$, for compact Hausdorff spaces $X$, i.e. all point evaluations on $C_c(X)$, as $\int fd\mu_p = f(p)$. This is not too hard to prove (I first saw it in Semadeni, Banach spaces of continuous functions). The proof I saw did not involve measures, though.
I think the same holds for $C_0(X)$, by considering $C_0(X)$ embedded into $C(\alpha(X))$ (where $\alpha(X) = X \cup \{\infty\}$ is the Alekandrov compactification of $X$), as all functions that vanish at $\infty$, and a character $\chi$ on $C_c(X)$ can be extended to one on $C(\alpha(X))$, using ideas like $\chi'(f) = \chi(f - f(\infty))$