What is the generating function for the sequence $\{kH_k\}_{k \geq 1}$
We know that a generating function for the Harmonic series is: 

$$
\sum_{i =1}^{\infty} H_i z^i = -\frac{\log(1-z)}{1-z}
$$

you may verify it by taking the MacLaurin expansion.

Taking the derivative with respect to $z$:

$$
\frac{d}{dz} \left ( \sum_{i =1}^{\infty} H_i z^i \right ) = \frac{d}{dz} \left ( -\frac{\log(1-z)}{1-z} \right )
$$

yields:
$$
\sum_{i = 1}^{\infty} iH_i z^{i-1} = \frac{1 -\log(1-z)}{(1-z)^2}
$$

Notice that LHS is exactly the formal series you mentioned.