First, let me expand slightly on the last paragraph of Geoff Robinson's answer. Your question is equivalent to the question whether $G=\text{GL}_2(\mathbb{F}_p)$ has a faithful 2-dimensional complex representation. Now, such a representation cannot be the sum of two one-dimensional ones, since they would all factor through G', so you are looking for an irreducible representation.
The irreducible representations of $\text{GL}_2(\mathbb{F}_p)$ are actually fairly easy to write down. If you induce all possible one-dimensional representations from $B=\begin{pmatrix}* & *\\0 & *\end{pmatrix}$, then the inductions will either be irreducible (hence $p+1$-dimensional) or they will be direct products of a one-dimensional and another irreducible. This way you get the characters lifted from G/G' and some $p$-dimensional irreducible ones. All the remaining irreducible representations of $G$ have dimension $p-1$ (they can also be written down explicitly, but slightly less easily). In summary, the dimensions of the irreducible representations of $G$ are 1, $p-1$, and $p$.
Now, to your motivation: usually when people try to "make mod $l$ representations complex", what they mean is lift them to characteristic 0. In other words, if $\overline{\rho}:\Gamma\longrightarrow \text{GL}_n(\mathbb{F}_l)$ is a representation, find a representation $ \rho:\Gamma\longrightarrow \text{GL}_n(\mathbb{Z}_l)\hookrightarrow \text{GL}_n(\mathbb{Q}_l) $ that reduces to $\overline{\rho}$ modulo $l$. When this can be done is a fairly delicate question, in general. After that, you can fix some embedding of $\mathbb{Q}_l$ into $\mathbb{C}$ if you want, to make $\rho$ into a complex representation.