Reducing directed hamiltonian cycle to graph coloring
Let $Q \in \sf{NP}$ and $Q&#39; \in \sf{NP\text{-}hard}$. Then, by definition, $Q$ is (many-one) reducible to $Q&#39;$ in polynomial time.

The exact chain of the reductions will depend on the $\sf{NP\text{-}hard}$ness proof of $Q&#39;$. Typically, it is proven by a chain of reductions starting from $\mathrm{SAT}$ and ending with $Q&#39;$ and then using the Cook-Levin theorem. So the chain of reductions will be a reduction from $Q$ to $\mathrm{SAT}$ followed by the chain of reductions from $\mathrm{SAT}$ to $Q&#39;$.

There is usually a more direct reduction for specific problems (without using Cook-Levin), since it is usually easy to find a propositional formula directly expressing the required property (with no reference to TMs). For example, in the case of Directed Hamiltionian Path ($\mathrm{DHP}$) and Graph Coloring ($\mathrm{GC}$), you can reduce:


$\mathrm{DHP}$ to $\mathrm{SAT}$,
$\mathrm{SAT}$ to $\mathrm{3SAT}$,
$\mathrm{3SAT}$ to $\mathrm{GC}$.