If we think of $\mathbb{C}^N$ as a commutative $C^*$-algebra, then it will be the algebra of continuous functions on $X=\{1,2, \dots, N\}$. So the only functions that are continuous are constants, (do we consider discrete topology in this case?).
Also, the product in an abstract commutative $C^*$-algebra $C(X)$ is defined pointwise as $fg(x) = f(x)g(x)$. How does this product change when I bring it to $\mathbb{C}^N$?
I know that it becomes the product of the coordinates of the two vectors pointwise, but I would like to figure better this connection.
You need to make this jump in your understanding: with $X=\{1,\ldots,n\}$, the equality $$\tag1 \mathbb C^n=\{f:X\to\mathbb C\} $$ is not something does needs to be proven but the definition of the left-hand-side.
I'm sure for years you have seen/writen $x\in\mathbb C^n$ and $x_2$ for the value of the second coordinate of $x$. Would it be clear if you write $x_2$ as $x(2)$? That's all that's happening in $(1)$.