Let $A$ be a graded ring, $M,N$ graded $A$-modules and $u:M\rightarrow N$ a graded linear map of degree $\delta$. I want to show that $\text{Im}(u)$ is a graded submodule of $N$. It is sufficient to prove that $\text{Im}(u)$ is generated by homogeneous elements.
Let $y\in\text{Im}(u)$. Then there exists $x\in M$ such that $y=u(x)$. We can write $y$ and $x$ as a unique sum of their homogeneous components: $x=\sum_{\lambda\in\Delta}m_\lambda$ and $y=\sum_{\lambda\in\Delta}n_\lambda$. This implies that $y=u(x)=\sum_{\lambda\in\Delta}u(m_\lambda)$, where $u(m_\lambda)\in N_{\lambda+\delta}$, for all $\lambda\in\Delta$.
I'm not sure what to do at this point; how can I use the above data to find a generating system of $N$ whose elements are homogeneous?
Notation: Let $(A_\lambda)_\lambda$, $(M_\lambda)_\lambda$ and $(N_\lambda)_\lambda$ be the gradings of $A,M$ and $N$, respectively.
You are basically done already. Each $u(m_\lambda)$ is a homogeneous element of $\text{Im}(u)$, so you have written $y$ as a sum of homogeneous elements of $\text{Im}(u)$. That is, $\text{Im}(u)$ is generated by its homogeneous elements.
(Note that you don't really need to find a generating set of homogeneous elements. You can simply take the set of all homogeneous elements of $\text{Im}(u)$, and see whether this set generates all of $\text{Im}(u)$.)