Let $J$ be an almost complex structure on an algebraic variety $V$. As we all know, $J$ comes from a complex structure if the Nijenhuis tensor of $J$ vanishes. What I would like to know is if there exists a simpler characterisation of integrability than this for varieties (as opposed to general manifolds).

If you want equivalent conditions to the Nijenhuis tensor vanishing then one is that the induced $\bar \partial$ operator defines a complex, i.e. that $\bar \partial^2 = 0$. Another one is that the exterior derivative decomposes as $d = \partial + \bar \partial$. If you want explicit examples of almost complex manifolds that are not complex, that's going to be more difficult, see the answers to tinyurl.com/4zrkar6 To find the $\bar \partial$ operator associated to an almost complex structure $J$ on a smooth manifold $M$, one needs to note that $J$ induces a splitting $T_M \otimes \mathbb C = T^{1,0} \oplus T^{0,1}$ of the tangent bundle into $i$ and $i$ eigenvectors (the eigenspaces being marked with $(1,0)$ and $(0,1)$, respectively). The same thing happens on the level of 1forms (and indeed on the level of $k$forms): they split into $(p,q)$forms like on complex manifolds. If $\pi^{p,q} : \bigwedge^k T_M \to \bigwedge^{p,q} T_M$ is the projection onto the space of $(p,q)$forms, then the $\bar \partial : \bigwedge^{p,q} T_M \to \bigwedge^{p,q+1} T_M$ operator associated to $J$ is $\bar \partial_J = \pi^{p,q+1} \circ d$. Once one does the calculations this comes out to $$ \bar \partial \alpha = \frac 1 2 \left( d \alpha + i d J \alpha \right) $$ for a $(p,q)$form $\alpha$. A similar formula holds for the $\partial$ operator, you just have to change $i$ to $i$. (I may have confounded the signs here.) A very good reference for the linear algebra parts (i.e. most of this) is Chapter 2 of Huybrecht's "Complex geometry". For the conditions equivalent to the vanishing of the Nijenhuis tensor I seem to remember the first chapter of http://tinyurl.com/4cqspu7 Moroianu's notes on Kahler geometry being very helpful when I went through this a couple of months ago. Finally, on why the vanishing of the Nijenhuis tensor implies that we indeed have a complex structure I recommend Demailly's book  Chapter 8, section 11 (page 396) has all the details: http://wwwfourier.ujfgrenoble.fr/~demailly/manuscripts/agbook.pdf 

