For ease of notation, I'll explain how to construct the nowhere-vanishing holomorphic volume form in the case $n = 2$, where the Calabi-Yau hypersurface is a quartic K3 surface. This method generalises for all $n$.
Let's first focus on the affine patch:
$$ U = \{ [1 : z_1 : z_2 : z_3] \ \vert \ (z_1, z_2, z_3) \in \mathbb C^3 \} \subset \mathbb{CP}^3.$$
On this affine patch $U$, the hypersurface is defined by a polynomial equation:
$$ f_{\rm quartic}(z_1, z_2, z_3) = 0.$$
For each $i \in \{1,2,3 \}$, we define the open subset
$$ V_i = \left\{ (z_1, z_2, z_3) \in \mathbb C^3 \ \big\vert \ \frac{\partial f_{\rm quartic}}{\partial z_i}(z_1, z_2, z_3) \neq 0\right\} \subset U.$$
Since $f_{\rm quartic}(z_1, z_2, z_3)$ is non-singular, we have
$$ U = V_1 \cup V_2, \cup V_3.$$
Now, observe that $z_2$ and $z_3$ form a set of holomorphic coordinates on $V_1$, by the holomorphic implicit function theorem (with $z_1$ determined holomorphically in terms of $z_2$ and $z_3$). Similar statements hold for $V_2$ and $V_3$.
We can therefore define non-vanishing holomorphic two-forms $\Omega_{V_1}$, $\Omega_{V_2}$ and $\Omega_{V_3}$ on $V_1, V_2, V_3$ respectively:
$$ \Omega_{V_1} = \frac{dz_2 \wedge dz_3}{\partial f_{\rm quartic}/\partial z_1}, \ \ \ \ \
\Omega_{V_2} = \frac{dz_3 \wedge dz_1}{\partial f_{\rm quartic}/\partial z_2}, \ \ \ \ \
\Omega_{V_3} = \frac{dz_1 \wedge dz_2}{\partial f_{\rm quartic}/\partial z_3},$$
It is easy to see that $$\Omega_{V_1} = \Omega_{V_2}$$ on $V_1 \cap V_2$. [One can verify this by multiplying the equation
$$ \frac{\partial f_{\rm quartic}}{\partial z_1} dz_1 + \frac{\partial f_{\rm quartic}}{\partial z_2} dz_2 + \frac{\partial f_{\rm quartic}}{\partial z_3} dz_3 = 0$$
by $\wedge dz_3$.]
Similarly, $\Omega_{V_2} = \Omega_{V_3}$ on $V_2 \cap V_3$, and $\Omega_{V_3} = \Omega_{V_1}$ on $V_3 \cap V_1$.
So $\Omega_{V_1}$, $\Omega_{V_2}$ and $\Omega_{V_3}$ glue together to define a non-vanishing holomorphic two-form $\Omega$ on the whole of the affine patch $U$.
There remains a possibility that $\Omega$ may vanish, or diverge, on the hyperplane
$$ H = \{ [0 : x_1 : x_2 : x_3] \ \vert \ [x_1 : x_2 : x_3] \in \mathbb{CP}^2 \}$$
which is not covered by the affine patch $U$. We must show that this does not happen.
Expressing this in the language of divisors, we know that ${\rm div}(\Omega) = nH$ for some $n \in \mathbb Z$, and our task is to show that $n = 0$. But this is obvious: since $\Omega$ is a meromorphic section of the canonical bundle, which is trivial for the quartic by adjunction, the divisor class ${\rm div}(\Omega)$ is the trivial divisor, hence $n = 0$. [Or if you want to avoid divisors, then just compute what $\Omega$ is after the change of coordinates and you'll see...]
As Gunnar pointed out, finding a Ricci-flat metric on the quartic K3 is a much more difficult problem. As far as I'm aware, my colleagues in string theory only know how to approximate this numerically.
Best Answer
In Métriques Kähleriennes et fibrés holomorphes, Calabi showed that the total space of the canonical bundle or the cotangent bundle of a positive Einstein-Kähler manifold admits a complete, Ricci-flat Kähler metric.
Corollary B.2 in A momentum construction for circle-invariant Kähler metrics extends this to the total space of a tensor product of canonical bundles over a product of positive Einstein-Kähler manifolds. The calculations here are explicit, enough so that it's easy to plot the fibre metric as a surface of rotation in Euclidean three-space.