Proving a statement about $k$-colouring of a graph 
Prove that a graph is $k$-colourable iff its edges can be oriented in such a way that the resulting directed graph does not contain a path of length $k$.

It seems to me that the '$\Leftarrow$' implication might be slightly easier to show if we were to consider directed acyclic graphs (then we could assign subsequent colors to vertices as we traverse along a path and do some rearranging when we visit an already coloured vertex)... But I am stuck at finding a valid argument that justifies the implication in either direction, not to mention I am actually interested in a stronger statement.

Perhaps you could use the Gallai-Hasse-Roy-Vitaver Theorem.
