What would the chemical name be for C13H8Cl3NO
Well, let's reconstruct that starting from the very right side, where it says a $\ce{C_6H_5}$. The ring and the $\ce{CO}$ group would be a benzaldehyde if it had an $\ce{H}$ instead of an $\ce{N}$, right? Or a benzoic acid if it was $\ce{OH}$ instead of $\ce{N}$. So what would it be if it had an $\ce{NH_2}$-group? It would be a benzamide. If the $\ce{N}$ is substituted with for example a chloride we call that N-chlorobenzamide. And now we have another $\ce{N}$-centered ligand, the second phenyl ring. The ring has three positions, the one where it's connected to the rest of the molecule would be 1, so that makes a 2,4-dichlorophenyl. 

Summarizing we get N-chloro-N-(2,4-dichlorophenyl)benzamide