As Hassan mentions, in the setting of singular metrics on vector bundles, the notion of curvature appears problematic, which is discussed in the paper of Raufi.

Still, as is also discussed in that paper, in the case that the singular metric is positively or negatively curved in the sense of Griffiths, one can give a reasonable meaning to $c_1(E,h)$ as a current by $c_1(E,h) := dd^c \log (\det h)$. This gives a way of defining the first Chern class without having defined a curvature matrix, but which coincides with the usual definition when $h$ is smooth.

Me, Raufi, Ruppenthal and Sera do a variant of this also for higher Chern classes, through local regularizations of the metric, under some additional assumption that the singularities of the metric are not too "big". This is so far not completely written up, but a discussion of it can be seen at
https://www.birs.ca/events/2016/5-day-workshops/16w5080/videos/watch/201605020920-Larkang.html

The precise statement of our main result is discussed at around 37:40, and basically, in the positively curved case, we can define $c_k(E,h)$ if the set $L(\log (\det h^*))$ is contained in a variety of codimension $\geq k$, where $L(\log(\det h^*))$ is the complement of the set where $\log(\det h^*)$ is locally bounded.

By taking the "diagonal" metric $h = e^{-\varphi} Id$ on a trivial vector bundle, where $\varphi$ is a plurisubharmonic function, one sees that one should expect to need some condition on the singularities of $h$, since $(dd^c \varphi)^k$ does typically not have a natural meaning when $\varphi$ is plurisubharmonic and unbounded, and $k$ is too large.