On an arbitrary almost complex manifold, $M$, it is said that one can always find coordinates for which the almost complex structure $J$ takes the canonical form $$ J_p=\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array}\right] $$ at any given point $p$. In general, however, it is not possible to find coordinates so that $J$ takes the canonical form on an entire neighborhood of $p$. Such coordinates, if they exist, are called 'local holomorphic coordinates for $J$'. If $M$ admits local holomorphic coordinates for $J$ around every point then these patch together to form a holomorphic atlas for $M$ giving it a complex structure, which moreover induces $J$. $J$ is then said to be 'integrable'.
My question is, how does one show that when local holomorphic coordinates exist, $J$ takes the canonical form in a local patch?
One need to distinguish complex structure on a manifold and almost complex structure on a manifold.
A complex structure is an atlas with holomorphic coordinates. An almost complex structure, however, is an automorphism of the tangent bundle $I$, such that $I^2 = -1$.
If I understood you correctly your question is: how to construct an almost complex structure from a complex structure on a manifold.
Well, then the answer is the following: let's do it locally. A neighborhood of a point $x \in M$ is identified with a complex vector space $\mathbb{C}^n$. That is the same as to identify $\mathbb{C}^n$ with the tangent space $T_xM$ in this point.
Choose any basis $e_1, \ldots, e_n$ in $\mathbb{C}^n$. Consider also vectors $f_j := ie_j$. the set $e_1, \ldots, e_n, ie_1, \ldots ie_n$ is a basis for $\mathbb{C}^n = T_xM$ viewed as a real vector space.
Define the operator $I$ as the following: $Ie_j = f_j$ and $If_j = -e_j$. This is an almost complex structure (since all the gluing maps on the charts are smooth, it dependes on the point smoothly).
Indeed the construction is rather simple, as well as the whole philosophy here: it is based on the idea that complex vector space is the same as an even-dimensional real vector space plus some operation of multiplication on the imaginary unity.