How to modify Bellman-Ford algorithm for this specific Minimum Cost Flow problem?
Minimum cost path

If you're looking for a minimum cost path, you can find that directly with Bellman-Ford.  Just let the weight on each edge $(i,j)$ be the cost $c_{ij}$, and ask Bellman-Ford to find for you the shortest path from the source to the sink; this will be exactly the minimum cost path (assuming there are no negative cycles in the graph).

Minimum cost flow

If you're looking for a minimum cost flow, one natural approach is to try using the Ford-Fulkerson method.  In each iteration of Ford-Fulkerson, you must find an augmenting path in the residual network.  In your particular case, you might try computing a minimum-cost augmenting path from the source to the sink.  Now what algorithm do you know that can compute a minimum-cost augmenting path in a digraph (potentially with cycles and potentially with negative-weight edges)?  That should be enough of a hint to give you something you can try.  Since it's your exercise problem, I'll let you work out the details and see if you can prove whether it works or not.

Alternatively, if conceptual simplicity is the goal, linear programming (LP) is one approach that is conceptually very simple and might apply to your porblem, and it has a polynomial running time.  LP immediately gives a solution to your problem (though possibly not with integer $x_{ij}$'s), and it's a well-known theorem that you can solve LP in polynomial time.  Of course, LP might be a "bigger hammer" than you want to throw at this.  Also, I don't know if LP will give you an integral flow, which you seem to want.  There's a standard theorem that says that, for the linear program associated with a maximum flow problem, there is always an integral solution to the LP that is optimal (you can't do better with non-integral flows).  I don't know if that carries over to the minimum-cost flow setting as well.