We can identify $\mathbb R / \mathbb Z$ with the circle group $\mathbb T=\{z \in \mathbb C \mid |z|=1\}$, with the multiplication coming from $\mathbb C$. We map $x \to e^{2 \pi i x}$. Then $x+y \pmod 1 \mapsto e^{2 \pi i (x+y) \pmod {2\pi}}$, so this will be a group isomorphism.
Then the characters of $\mathbb T$ are the continuous group homomorphisms from $\mathbb T \to \mathbb T$. So, thinking about this geometrically, we start at $1$ and then wind our way around the circle an integer number of times. It must be an integer because group homomorphisms must map $1$ to $1$. Anticlockwise or clockwise will correspond to positive and negative integers. Clearly, each integer will give us a different character and we obtain all characters in this way.
So the set of characters is given by $ \{x \mapsto e^{2 \pi imx} \mid m \in \mathbb Z \}, \text{ for } x \in \mathbb R/\mathbb Z. $