To prove that every complex manifold $M$ is an almost complex manifold, it suffices to show that $J_p:T_pM\rightarrow T_pM $ defined by $J_p(\partial/\partial x_i)=\partial/\partial y_i$ and $J_p(\partial/\partial y_i)=-\partial/\partial x_i$ is independent of the choice of the chart around $p\in M$.
But why is this sufficient? Don't we need to prove that $J:TM\rightarrow TM$ is smooth? How does independence of the choice of charts imply smoothness of $J$?
Given a complex chart $z_i = x_i + \sqrt{-1}y_i$ defined on a neighborhood $U$ of $M$, you can define $J$ on $U$ by the formula you wrote. Then $J$ is represented with respect to the smooth frame $\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1}, \dots, \frac{\partial}{\partial y_n}$ of $TM$ (over $U$) by a matrix whose entries are smooth (even constant) and so $J$ is smooth on $U$.
If you show that this definition is independent of the chart, you can use it to define $J$ globally and it will be smooth because it is locally smooth.