In your context, $\ce{C'}$ is the carbon atom in the carbonyl ($\ce{C=O}$) group. However, I would add that $\ce{C'}$ doesn't appear like a standard notation. In organic chemistry, the alpha carbon is the carbon immediately adjacent to a functional group. The $\ce{C_\alpha}$ in the case of amino acids is alpha carbon with respect to both the amine and carbonyl.

In your context, the "plain-C" labelled carbon is the carbon atom in the carbonyl ($\ce{C=O}$) group. In organic chemistry, the alpha carbon is the carbon immediately adjacent to a functional group. In general, the greek letters in alphabetical order denote the distance in increasing order from a functional group. The $\ce{C_\alpha}$ in the case of amino acids is alpha carbon with respect to both the amine and carbonyl.