Pach and Toth [proved](http://www.renyi.hu/~pach/publications/fewcrossing.pdf) that if $G$ is a $k$-planar graph (with $k \geq 1$), then 

$|E(G)| \leq 4.108 \sqrt{k} |V(G)|$.

Thus, every $k$-planar graph has a vertex of degree at most $\lfloor8.216 \sqrt{k}\rfloor$. By induction, it follows that $k$-planar graphs can be coloured with $\lfloor8.216 \sqrt{k}\rfloor+1$ colours.  

For $1 \leq k \leq 4$, they prove a better bound of 

$|E(G)| \leq (k+3)(|V(G)|-2)$. 

Thus, we can do slightly better for small values of $k$.  That is, 2-planar graphs are 10-colourable, 3-planar graphs are 12-colourable, and 4-planar graphs are 14-colourable.