I am not sure how to separate function to apply chain rule for $\sum_{k}(\sum_{j}\vert H_{kj}\vert)^{2}$ wrt. to $H_{kj}$ where $H_{kj}$ is a matrix. What I have tried so far is:
$$g(h) = \sum_{j}\vert h_{j}\vert$$
$$f(x) = x^2$$
I omit $\sum_{k}$ first. Then, $(\sum_{j}\vert H_{kj}\vert)^{2}$ can be written as $f(g(H_{kj}))$:
\begin{equation}\begin{aligned} f(g(H_{kj}))' &= f'(g(H_{kj})) \times g'(H_{kj}) \\ &= 2g(H_{kj}) \times \sum_{j}1 \\ &= 2\sum_{j}\vert H_{kj}\vert \times \sum_{j}1 \\ &= 2\sum_{j}\vert H_{kj}\vert \times j \end{aligned}\end{equation}
I think $\sum_{j}1$ should be sum of $1$, $j$ times but it seems weird. Is the final result $2j\sum_{kj}\vert H_{kj}\vert$ or $2\sum_{kj}\vert H_{kj}\vert$?
Instead of the chain rule, you can solve the problem using differentials and standard matrix notations.
For the elementwise/Hadamard and trace/Frobenius products we'll use the symbols $$\eqalign{&A\odot B\cr &A:B}$$ respectively. Note that $(A:B)$ is defined as $\,{\rm Tr}(A^TB)$
Next, define the elementwise sign function $\,\,S={\rm sign}(H)$
$$\eqalign{ S_{ij} &= \begin{cases} +1 &{\rm if\,\,\,\,} H_{ij}>0 \\ \,\,\,\,\,0 &{\rm if\,\,\,\,} H_{ij}=0 \\ -1 &{\rm if\,\,\,\,} H_{ij}<0 \\ \end{cases} \cr }$$ The S-matrix allows us to write $\,\,\,|H|=S\odot H$
Finally, we'll use $u$ to denote the vector whose elements are all unity, so that summation over an index can be replaced by a matrix product with the $u$ vector, e.g. $\,\,Au = \sum_kA_{ik}$
In terms of these definitions, your function can be written in matrix notation $$\eqalign{ \phi &= u:\Big((S\odot H)u\Big)\odot\Big((S\odot H)u\Big) \cr }$$ To reduce some clutter, let's temporarily define a vector $\,\,x=(S\odot H)u$
and find the differential and gradient of the function. $$\eqalign{ \phi &= u:(x\odot x) \cr d\phi &= u:(2x\odot dx) = 2x:dx = 2x:(S\odot dH)u = 2(S\odot xu^T):dH \cr \frac{\partial\phi}{\partial H} &= 2S\odot xu^T = 2S\odot\Big((S\odot H)uu^T\Big) = 2S\odot\Big(\,|H|\,uu^T\Big) \cr \cr }$$ Note that $uu^T$ is a square matrix whose elements are all unity.