Ciao Francesco! 

In general if a projective manifold $X$ has ample canonical bundle, then it admits a Kähler-Einstein metric $\omega$ of negative Einstein constant, *i.e.* a Kähler metric such that $\operatorname{Ric}(\omega)=-\omega$ (by the Aubin-Yau Theorem). 

In particular $T_X$ admits a Hermite-Einstein metric, and then so does any irreducible $\operatorname{GL}(T_X)$-representation, such as the symmetric powers of the cotangent bundle. This makes these vector bundles  $[\omega]$-semistable, by the easy direction of the Kobayashi-Hitchin correspondence.

Finally, in this case, begin $[\omega]$-semistable means that these vector bundle are $K_X$-semistable, since $[\omega]=[-\operatorname{Ric}(\omega)]=c_1(K_X)$.

You can see all this and much more on S. Kobayashi "Differential geometry of complex vector bundles".

P.S. The work of Bogomolov you are looking for is 

"Holomorphic tensors and vector bundles on projective varieties", Math. USSR Izvestija 13/3 (1979) 499-555.

For surfaces you don't need $K_S$ to be ample, big&nef suffices.