My understanding of a dual space is that it is a space of all the linear bounded operators that transform objects in Sobolev space (for this case) to the real or complex spaces. I am not sure if this is complete or even correct. Also, I don't really get what the Cauchy-Schwarz inequality has to do with this.
To solve this problem you need to know a little bit about Sobolev space theory, the definition of $H^1$ and its norm. By the Cauchy-Schwarz inequality
$$| \ell(v)| \leq \left( \int_{\Omega} |f|^2 dx \right)^{1/2} \left( \int_{\Omega} |v|^2 dx \right)^{1/2} = ||f||_2 ||v||_2 \leq ||f||_2 ( ||v||_2 + C||Dv||_2)= ||f||_2 K||v||_{V}.$$
Then, by definition of operator norm $$ || \ell|| \leq K||f||_2 $$