The correction term for Durand-Kerner root finding method is
$w_k = -\frac{f(z_k)}{\prod_{j\not=k}(z_k - z_j)}$
Wikipedia Talk page mentions that it is also possible to use derivative in denominator, instead of the above product.
How to form such derivative? All I have are coefficients of the polynomial and the approximations of the roots. How to come up with coefficients of the derivative so I can evaluate it with Horner scheme like I do it for evaluation of polynomial (the $f(z_k)$)?
Am I correctly assuming that the derivative is looks perhaps like $g'(x)$ where $g(x) = \prod(z_k - z_j)$?
PS: I tried to implement the expression by Bo Jacoby from the Talk page but I cannot get it to work: I tried to sum all the products of all the approximations but one and put the result into denominator but it doesn't seem to work that way...
Yes, you assume correctly that the update can also be written as $$ w_k=-\frac{f(z_k)}{g'(z_k)}. $$
And correct, the derivative of $g$ can be written as sum of products according to its construction as product of linear factors $$ g(x)=(x-z_1)…(x-z_n) $$ (note that the leading coefficient of $g$ is $1$, so that this must also hold for $f$) $$ g'(x)=\sum_{k=1}^n\;\prod_{j\ne k}(x-z_j). $$
However, for small to medium degrees ($\deg f < 70$ or so) there is no gain in going the route via the construction of $g$ and $g'$. The standard evaluation of the products and of the value of $g'$ have about the same number of operations, and in the second case there is the overhead of constructing $g$.
For larger degrees one can use Karatsuba or FFT based methods to speed up the multi-point evaluation of $g'$ so that this way becomes faster. (But then the question is of how viable numerical results in general are due to the high degree.)