weighted-clique to vertex cover reduction
Let $G=(V,E)$ be an unweighted graph (think of all the edges as having weight $1$), $n=|V|$, and $\bar{G}$ be the edge-complement of $G$.

Now, $C$ is a vertex cover of $G$ iff $\bar{C} = V \setminus C$ is an independent set of $G$, that is, iff $\bar{C}$ is a clique of $\bar{G}$.

When you combine this with the fact that a clique of size $k$ has $\binom{k}{2} = \frac{k^2-k}{2}$ edges you get: "There is a vertex cover of size $k$ in $G$ iff $\bar{G}$ contains a clique of size $a = n-k$ and weight $b = \binom{n-k}{2}$." 

In fact, as far as this reduction is concerned, you can drop either the size or the weight constraint from your clique problem since one implies the other.