even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.

Let us consider a smooth projective algebraic variety $X$ over $\mathbb{C}$. Whenever I have a coherent sheaf $F$ over $X$, I can get a resolution of $F$ by a (bounded) complex $E^\bullet$ of vector bundles on $X$ and define $\mathrm{ch}(F):=\sum (-1)^i\mathrm{ch}(E^i)$. Since I (should) know how to compute Chern classes of vector bundles via splitting principle, this makes sense. I can indeed already start with a complex of coherent sheaves $F^\bullet$ on $X$, replace by vector bundles and do the exact same thing, defining a map $\mathrm{ch}\colon D^b(CohX)\to H^\ast (X,\mathbb{Q})$, or $CH^\ast(X)$, or $HH$ something, depending on your taste (I stick to the cohomology one), which actually factors through $K(X)$ because of additivity of $\mathrm{ch}$ with respect to short exact sequences.

The other good point of vector bundles is that we have an intuition of what Chern classes are, that is some kind of objects which measure wheter certain number of generic sections of our bundle are linearly dependent.

I am now trying to reconciliate the intuition with the homological nonsense, and I would like to have some example of what can we know about some sheaf when we know its Chern character.

A precious one for me would be the possibility to characterise torsion sheaves on $X$ (e.g. $\mathcal{O_x}$ skyscrapers, $\mathcal{O_Z}$ structure sheaf of a curve/subvariety) K3 surface.

Thank you for your time!