To solve explicitly this system, one could use the change of variables $z=xy$ to get the equivalent system $z'=(b+c)z+a+d+Cx$ and $x'=x(b+a/z)$, from which one can deduce a differential equation in $z$ only.
But to study the stability of a differential system $x'=f(x,y)$, $y'=g(x,y)$, one does not need to solve it. Instead, a phase diagram is often sufficient. The idea is to consider the plane $(x,y)$ and to point an arrow at each point $(x,y)$ in the direction of $(f(x,y),g(x,y))$. The path $t\mapsto (x(t),y(t))$ followed by a solution is then visible and its asymptotics (involving the fixed points of the system) may become obvious.
In your case, much will depend on the signs of the parameters $a$, $b$, $c$, $d$ and $C$, and maybe of the initial point $(x(0),y(0))$. My guess is that you are considering positive solutions hence useful conditions might be that $a$, $d$ and $C$ are positive.
For an example and for some more detailed explanations, see here.