If you want to clip all the way through your polydata with a cylinder, then the fastest way is to use vtkClipPolyData with vtkCylinder. The vtkCylinder defines an implicit function for the cylinder, instead of defining the cylinder with polygons. Clipping polydata with a function is hundreds of times faster than clipping polydata with another polydata.
For example, see this example (unfortunately, it’s not a good example, since the example is much more complicated than what you need).
Edit: Note that if you use vtkImplicitDataSet to create the implicit function, you lose all of the speed benefits of implicit functions. The fast implicit functions are the simple ones: vtkPlane, vtkCylinder, vtkSphere and vtkImplicitVolume.