How do I Program that?
Single Deformable Mesh
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• Each bone has a list of vertices it influences and to what degree.

• All weight on a single vertex add up to 100%.

• For each vertex, calculate where it would have been if it were 100% under the influence of each bone.

• Take that delta and multiply it by the weighting factor.

• Add the deltas up for all the influencing bones and that is your final position in object space.

• In other words:
  For each vertex V
  
  For each bone B that influences V
  Delta[B] = Pos[V] * Matrix[B] - Pos[V]
  FinalPos[V] += Delta[B] * weight[B]

  One tricky thing is that the mesh needs to be offset so that when it is rotated by the matrix, it rotates about the bones axis.