Let $p$ be a prime number. Denote by $G$ the absolute Galois group of (a finite extension of) $\mathbb{Q}_p$. Let $\ell$ be a prime number.
For $\ell= p$, I guess it is well known that the irreducible continuous and finite dimensional representations of $G$ with coefficients in $\bar{\mathbb{F}}_p$ are induced from a character. More precisely, let $d$ be the dimension of the representation which we denote by $\rho$ and let $\mathbb{Q}_{p^d}$ be the unramified extension of $\mathbb{Q}_p$ of degree $d$, with absolute Galois group $G_d$. Then there exist a character $\omega : G_d \to \bar{\mathbb{F}}_p^{\times}$ such that $\rho$ is isomorphic to $Ind_{G_d}^{G} (\omega)$ (and I think $\omega$ is a power of a fundamental character of "niveau" $d$ as defined by Serre).
Now can we describe the irreducible representations of $G$ with coeffients in $\bar{\mathbb{F}}_{\ell}$ when $\ell \neq p$ ?