From the binomial theorem,
$\displaystyle (1+x)^k=\sum_{j=0}^{k} {{}^{k}\textrm{C}_{j}}~x^j$
Putting $x=-1$ to equation above yields,
$\displaystyle (1-1)^k=\sum_{j=0}^{k} {{}^{k}\textrm{C}_{j}}(-1)^j\\ \displaystyle \Rightarrow \sum_{j=0}^{k} (-1)^j~{{}^{k}\textrm{C}_{j}}=0$
Now
$\displaystyle \sum_{k=1}^{n}(-1)^k~{{}^{n}\textrm{C}_{k}} \sum_{j=0}^{k} (-1)^j~{{}^{k}\textrm{C}_{j}}=\sum_{k=1}^{n}(-1)^k~{{}^{n}\textrm{C}_{k}} \times 0=0$
So the correct answer is B.