  We consider non-Hermitian random matrices $X \in \mathbb{C}^{n \times n}$
with general decaying correlations between their entries. For large $n$, the
empirical spectral distribution is well approximated by a deterministic
density, expressed in terms of the solution to a system of two coupled
non-linear $n \times n$ matrix equations. This density is interpreted as the
Brown measure of a linear combination of free circular elements with matrix
coefficients on a non-commutative probability space. It is radially symmetric,
real analytic in the radial variable and strictly positive on a disk around the
origin in the complex plane with a discontinuous drop to zero at the edge. The
radius of the disk is given explicitly in terms of the covariances of the
entries of $X$. We show convergence down to local spectral scales just slightly
above the typical eigenvalue spacing with an optimal rate of convergence.
