Two alternating Euler sums

Question

How can we evaluate the following series: $$\sum\limits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right){{\left( {\sum\limits_{k = 1}^n {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{k}} } \right)}^2}} ,\sum\limits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{{{n^2}}}\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right){{\left( {\sum\limits_{k = 1}^n {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{k}} } \right)}^2}}$$