To elaborate on Emerton's answer, an arbitrary (finitely ramified and odd) representation $\rho_n:G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{Z}/p^n\mathbb{Z})$ can't lift to one coming from an eigenform because of some reasons which may be explained locally. Let $\bar{\rho}$ denote the mod $p$ representation and $Ad^0\bar{\rho}$ the $\mathbb{F}_p[G_{\mathbb{Q}}]$-module of trace zero $2\times 2$ matrices over $\mathbb{F}_p$ equipped with the adjoint action, i.e.$g\in G_{\mathbb{Q}}$ acts on a matrix $A$ by conjugation $g\cdot A:=\bar{\rho}(g) A\bar{\rho}(g)^{-1}$. Let $l$ be a prime at which $H^2(G_{\mathbb{Q}_l}, Ad^0\bar{\rho})\neq 0$ (here $G_{\mathbb{Q}_l}$ is a decomposition subgroup at $l$). The obstruction-class associated with ${\rho_n}_{\restriction G_{\mathbb{Q}_l}}$ is a cohomology class $O_l({\rho_n}_{\restriction G_{\mathbb{Q}_l}})\in H^2(G_{\mathbb{Q}_l}, Ad^0\bar{\rho})$ which is non-zero if there is no lift of the local representation ${\rho_n}_{\restriction G_{\mathbb{Q}_l}}$ to $GL_2(\mathbb{Z}/p^{n+1}\mathbb{Z})$. This local obstructedness phenomenon is an issue when $n\geq 2$ though not so for $n=1$. This can be further explained as follows, at each prime $l$ there is a smooth subscheme in the scheme parametrizing local deformations of $\bar{\rho}_{\restriction G_{\mathbb{Q}_l}}$, if for $n\geq 2$ at some prime $l$ where $H^2(G_{\mathbb{Q}_l}, Ad^0\bar{\rho})\neq 0$ it is possible that ${\rho_n}_{\restriction G_{\mathbb{Q}_l}}$ does not lie on this smooth subscheme in the space of all deformations of ${\rho_n}_{\restriction G_{\mathbb{Q}_l}}$. In this case, ${\rho_n}_{\restriction G_{\mathbb{Q}_l}}$ cannot lift one more step (and thus the global representation $\rho_n$ will not lift one more step either). The construction of these smooth schemes in the spaces of local deformations (dubbed smooth/liftable local deformation conditions) is carried out in the paper of Ramakrishna: "Deforming Galois representations and the Conjectures of Serre and Fontaine-Mazur" and at the prime $p$ this was previously carried out in his paper "On a Variation of Mazur's Deformation Functor".

There is now another issue, one does require that the lift to characteristic zero must satisfy a $p$ adic Hodge theoretic condition at $p$, this can be done when $n=1$ but there can be issues when $n\geq 2$.

It would be of interest to know if in case the local representations ${\rho_n}_{\restriction G_{\mathbb{Q}_l}}$ are all unobstructed (i.e. their obstruction classes are trivial, i.e. they lift to $GL_2(\mathbb{Z}/p^{n+1}\mathbb{Z})$)) at the finitely many primes at which it is unramified, and if $\rho_{\restriction G_{\mathbb{Q}_p}}$ satisfies a further condition, then if indeed it lifts to the representation associated to a cuspidal eigenform/ big Hecke algebra?