How to implement the details of shotgun hill climbing to make it effective?
The general idea behind doing multiple climbs is to try to avoid local optima. This is the reason shotgun climbing might work better than just a plain hill climbing method.

Usually one starts with a random value. On the other hand if one can guess something better, that can be used as a initial value as well. If there is no knowledge of what a good start value should be, it makes sense to use a random one.

The suitable value for $N$ varies. Usually one has to define a stopping criterion. A suitable criterion might depend on your specific domain. For example, if for some small number of moves you don't improve your solution at all, you stop. In general, you don't know -- you are using heuristics and they give you no guarantees. You might want to even determine a suitable $N$ experimentally.