Let $X$ be a matrix representation. Let the kernel of $X$ be defined as $N = {\{g \in G: X(g) = I}\}$. A representation is faithful if it's one to one.
Show that $N$ is a normal subgroup of $G$ and find a condition on $N$ equivalent to the representation being faithful.
Proof:
Let $X : G → GL(V)$ be a group representation. Let $g_1 \in N$ and $g \in G$.
Then $$X(g^{-1}g_1g) = X(g^{-1})X(g_1)X(g) = X(g)^{-1}(I)X(g) = X(g)^{-1}X(g) = I.$$
Thus $g^{-1}g_1g \in N$, so $N$ is a normal subgroup of $G$.
Further, $X$ is faithful if and only if $N$ is the identity subgroup of $G$.
Can someone please verify, or give feedback on, this proof.
Yes, everything is in working order here. It's the same format to prove that the kernel of any homomorphism is a normal subgroup, so I'm a little surprised you didn't just say that the representation $X$ is, among other things, a homomorphism and thus its kernel is a normal subgroup of $G$.
You also have a perfectly good characterization of a faithful representation, that it has a trivial kernel. The non representation-specific version is that any homomorphism is injective if and only if its kernel is trivial.