Disprove for a space $X$ a property P such that all metric spaces have property P.
Such P include $X$ is Hausdorff, $X$ is normal, $X$ is perfectly normal, $X$ is first countable.
Also for metric spaces we have that
being ccc, separable, Lindelöf, second countable are all equivalent (if $X$ is metrisable and has one of these properties, it has all the other ones). If for some $X$ we have that e.g. $X$ is separable but not second countable (as for the Sorgenfrey line (aka the lower limit topology)) then $X$ cannot be metrisable.
For metric spaces we also have that countably compact, pseudocompact, compact
and sequentially compact are all equivalent, so here the same strategy can be applied. e.g. $\omega_1$ is countably compact but not compact, so cannot be metrisable.
I think most proofs of non-metrisability are done along these lines.
A totally general method is not really possible I think.