As far as I know no one has “written down” a hyperkähler structure on a K3 surface. Indeed much of the work in Mirror symmetry revolves around trying to give an asymptotic expansion of such metrics (in this and other cases) in terms of curve counts and the like on the dual. Yau’s solution of the Calabi conjecture implies that every K3 surface admits a unique such structure.  The holomorphic 2-form is then parallel. This together with the Kähler form (also parallel) implies that the bundle of self dual forms is trivial. For any self dual form $\omega$ on an oriented Riemannian four-manifold we have a map 
$$
J_\omega:T^*M \to T^*M
$$
given by
$$
J_\omega(\theta)=\star\omega \wedge \theta
$$
where $\star$ is the Hodge star. 
One checks that 
$$
J_\omega^2=-2|\omega|^2.
$$
The sphere of self-dual  2-forms of length $1/\sqrt{2}$ acts as almost complex structures.  For our Yau metric this 2-sphere bundle admits a parallel trivializations and this gives the action of the imaginary quaternions.