Graph coloring/partitioning problem
Your problem is NP-complete.
Take an instance $G'=(V',E')$ of the (NP-complete) max-cut problem. Let $m=|E'|$.


Create two new vertices $x_1$ and $x_2$, create the edge between $x_1$ and $x_2$, and all edges between $V'$ and $\{x_1,x_2\}$.
The edge $[x_1,x_2]$ receives positive weight $1000m$
All edges $[x_i,v]$ with $i=1,2$ and $v\in V'$ receive negative weight $-2m$
All edges in $E'$ receive positive weight $1$


It can be easily seen that your optimal coloring for this graph


assigns two distinct colors to $x_1$ and $x_2$ (say color 1 to $x_1$ and color 2 to $x_2$)
and colors every vertex $v\in V'$ with one of these colors 1 or 2


Hence, the two-coloring on $V'$ induces a max-cut in $G'$.