A $1$-dimensional representation of a group is a continuous homomorphism into $\Bbb C^\times$: in particular it is equal to its own trace (once we identify $\mathrm{GL}_1(\Bbb C)\cong\Bbb C^\times$). So the group homomorphism definition of characters is the $1$-dimensional case of the trace-of-a-representation definition. To copy myself:
The homomorphisms $G\to \Bbb C^\times$ are actually not the whole story of character theory, but are a very tidy chapter in it. If $V$ is a vector space (over $\Bbb C$) and $G$ finite, the homomorphisms $G\to GL(V)$ from $G$ into the general linear group of invertible linear maps are called representations, which are essentially the ways to equip $V$ with a linear $G$ action. If $\rho$ is a representation, then the map given by $\chi_\rho:G\to \Bbb C:g\mapsto\mathrm{tr}\,\rho(g)$ (the trace of the linear map associated to $g$, which is independent of basis or coordinate choice for $V$) is called a character.
If $V$ is one-dimensional (in which case we call $\rho$ and $\chi_\rho$ one-dimensional as well) then $\rho=\chi_\rho$ and the characters are multiplicative. Note that $\mathrm{tr}\,\rho(e_G)=\dim\,V$ shows the dimension can be directly computed from the character, so there is no ambiguity with respect to what dimension a character may have. With a distinguished basis we have $V\cong \Bbb C^n$ in an obvious way, so we can write $GL(V)$ as $GL_n(\Bbb C)$, in which we are working with matrix representations specifically.
(Over here.)