Typically, you should prune in the winter, and while pruning itself is fairly easy, it's still a delicate procedure. Firstly, prune the shoots on the main stem. Then, any dead branches, or any that are rubbing, as that can create raw spots on the tree. These you should cut back to the parent branch, and leave only a stub. Then, every 4-5 years prune the branches around 4/5 of the way back to the main stem.

It is recommended that Kilmarnock willows be pruned in the winter. How you prune it is up to you, depending on the shape you want. Just don't cut off so many branches that the tree will not recover.