Your forcing notion will collapse all uncountable cardinals
below $\kappa$ to $\aleph_1$. To see this, fix any
uncountable $\gamma\lt\kappa$. Consider the first
$\aleph_1$ many cardinals after $\gamma$ at which forcing
occurs. For the $\xi^{th}$ such cardinal $\alpha$, we are adding a
subset to $\alpha^{++}$. Consider the first ordinal in the set
added at this stage. This can be any ordinal up to
$\alpha^{++}$, including any ordinal below $\gamma$. Because you are using countable support, any condition specifies nontrivial sets only on a countable number of these cardinals, and so it is dense that any particular $\beta\lt\gamma$ appears at least
once in such a way. Thus, in the generic extension, there
will be a surjective map from $\aleph_1$ onto $\gamma$, and
so $\gamma$ is collapsed.

The cardinal $\kappa$ itself is not collapsed, by a
$\Delta$-system argument (and neither is any cardinal above
$\kappa$ collapsed), and so this forcing makes $\kappa$ the
$\aleph_2$ of the extension. In particular, it does not
preserve the inaccessibility of $\kappa$.